Proof.
The implication $(\ast '_ n) \Rightarrow (\ast ''_{n} )$ is immediate. We will prove the converse. Assume that $(\ast ''_ n)$ is satisfied, and let $g: \operatorname{\partial \Delta }^ n \rightarrow \operatorname{\mathcal{C}}$ be an arbitrary morphism of simplicial sets; we wish to show that $g$ can be extended to a morphism $\overline{g}: ( \operatorname{\partial \Delta }^{n} ) \rightarrow \operatorname{\mathcal{C}}$. If $n=1$, this follows immediately form $(\ast ''_ n)$; we will therefore assume that $n \geq 2$. Note that we can write $\operatorname{\partial \Delta }^{n}$ as the union of $\Delta ^{n-1}$ and the horn $\Lambda ^{n}_{n}$, whose intersection is the simplicial subset $\operatorname{\partial \Delta }^{n-1} \subset \Delta ^{n-1}$. Set
\[ g_{-} = g|_{ \Delta ^{n-1} } \quad \quad g_{\pm } = g|_{ \operatorname{\partial \Delta }^{n-1} } \quad \quad g_{+} = g|_{ \Lambda ^{n}_{n} }. \]
Let $X = g(0)$ and $Y = g(n)$ and let $\pi : \operatorname{\mathcal{C}}_{/Y} \rightarrow \operatorname{\mathcal{C}}$ denote the projection map, so that we can identify $g_{+}$ with a morphism $\widetilde{g}_{\pm }: \operatorname{\partial \Delta }^{n-1} \rightarrow \operatorname{\mathcal{C}}_{/Y}$ satisfying $\pi \circ \widetilde{g}_{\pm } = g_{\pm }$.
Let $f_{-}: \Delta ^{n-1} \rightarrow \operatorname{\mathcal{C}}$ be the constant morphism taking the value $X$, and let $h_{-}: f_{-} \rightarrow g_{-}$ be the natural transformation given by the composite map
\[ \Delta ^{1} \times \Delta ^{n-1} \xrightarrow {(i,j) \mapsto ij} \Delta ^{n-1} \xrightarrow { g_{-} } \operatorname{\mathcal{C}}. \]
Set $f_{\pm } = f_{-} |_{ \operatorname{\partial \Delta }^{n-1} }$ and $h_{\pm } = h_{-} |_{ \Delta ^1 \times \operatorname{\partial \Delta }^{n-1} }$, so that $h_{\pm }$ can be regarded as a natural transformation from $f_{\pm }$ to $g_{\pm }$. Since $\pi $ is a right fibration, we can lift $h_{\pm }$ to a natural transformation $\widetilde{h}_{\pm }: \widetilde{f}_{\pm } \rightarrow \widetilde{g}_{\pm }$ in the $\infty $-category $\operatorname{Fun}( \operatorname{\partial \Delta }^{n-1}, \operatorname{\mathcal{C}}_{/Y} )$. Let us identify $\widetilde{f}_{\pm }$ with a morphism of simplicial sets $f_{+}: \Lambda ^{n}_{n} \rightarrow \operatorname{\mathcal{C}}$ satisfying $f_{+}(n) = Y$. Then $\widetilde{h}_{\pm }$ determines a natural transformation $h_{+}: f_{+} \rightarrow g_{+}$, given by the composition
\[ \Delta ^1 \times \Lambda ^{n}_{n} \simeq \Delta ^1 \times (\operatorname{\partial \Delta }^{n-1} )^{\triangleright } \rightarrow ( \Delta ^1 \times \operatorname{\partial \Delta }^{n-1} )^{\triangleright } \xrightarrow { \widetilde{h}_{\pm } } (\operatorname{\mathcal{C}}_{/Y})^{\triangleright } \rightarrow \operatorname{\mathcal{C}}. \]
Note that $f_{-}$ and $f_{+}$ can be amalgamated to a morphism $f: \operatorname{\partial \Delta }^{n} \rightarrow \operatorname{\mathcal{C}}$, and that $h_{-}$ and $h_{+}$ can be amalgamated to a natural transformation $h: f \rightarrow g$ in $\operatorname{Fun}( \operatorname{\partial \Delta }^{n}, \operatorname{\mathcal{C}})$.
Invoking hypothesis $(\ast ''_{n})$, we see that $f$ can be extended to a morphism $\overline{f}: ( \operatorname{\partial \Delta }^{n} )^{\triangleright } \rightarrow \operatorname{\mathcal{C}}$. Let $Z \in \operatorname{\mathcal{C}}$ denote the image under $\overline{f}$ of the cone point and let $\varphi : \operatorname{\mathcal{C}}_{/Z} \rightarrow \operatorname{\mathcal{C}}$ denote the projection map, so that $\overline{f}$ can be identified with a morphism of simplicial sets $f': \operatorname{\partial \Delta }^{n} \rightarrow \operatorname{\mathcal{C}}_{/Z}$ satisfying $\varphi \circ f' = f$. Let us identify the vertex $f'(n) \in \operatorname{\mathcal{C}}_{/Z}$ with a morphism $v: Y \rightarrow Z$ in the $\infty $-category $\operatorname{\mathcal{C}}$, so that we have a commutative diagram
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}_{/v} \ar [r]^-{\varphi '} \ar [d]^{\pi '} & \operatorname{\mathcal{C}}_{/Y} \ar [d]^{\pi } \\ \operatorname{\mathcal{C}}_{/Z} \ar [r]^-{ \varphi } & \operatorname{\mathcal{C}}. } \]
Set $f'_{+} = f'|_{ \Lambda ^{n}_{n} }$ and $f'_{\pm } = f'|_{ \operatorname{\partial \Delta }^{n-1} }$, so that we can identify $f'_{+}$ with a morphism $\widetilde{f}'_{\pm }: \operatorname{\partial \Delta }^{n-1} \rightarrow \operatorname{\mathcal{C}}_{/v}$ satisfying $\pi ' \circ \widetilde{f}'_{\pm } = f'_{+}$. Since the inclusion $\{ 0\} \hookrightarrow \Delta ^{1}$ is left anodyne, the morphism $\varphi ': \operatorname{\mathcal{C}}_{/v} \rightarrow \operatorname{\mathcal{C}}_{/Y}$ is a trivial Kan fibration (Corollary 4.3.6.13). We can therefore lift $\widetilde{h}_{\pm }$ to a natural transformation $\widetilde{h}'_{\pm }: \widetilde{f}'_{\pm } \rightarrow \widetilde{g}'_{\pm }$ for some morphism $\widetilde{g}'_{\pm }: \operatorname{\partial \Delta }^{n-1} \rightarrow \operatorname{\mathcal{C}}_{/v}$. Let us identify $\widetilde{g}'_{\pm }$ with a morphism $g'_{+}: \Lambda ^{n}_{n} \rightarrow \operatorname{\mathcal{C}}_{/Z}$ satisfying $\varphi \circ g'_{+} = g_{+}$. Then $\widetilde{h}'_{\pm }$ determines a natural transformation $h'_{+}: f'_{+} \rightarrow g'_{+}$, given by the composition
\[ \Delta ^1 \times \Lambda ^{n}_{n} \simeq \Delta ^1 \times (\operatorname{\partial \Delta }^{n-1} )^{\triangleright } \rightarrow ( \Delta ^1 \times \operatorname{\partial \Delta }^{n-1} )^{\triangleright } \xrightarrow { \widetilde{h}'_{\pm } } (\operatorname{\mathcal{C}}_{/v})^{\triangleright } \rightarrow \operatorname{\mathcal{C}}_{/Z}. \]
Let $e$ denote the restriction $h'_{+} |_{ \Delta ^1 \times \{ 0\} }$, which we regard as an edge of the simplicial set $\operatorname{\mathcal{C}}_{/Z}$. By construction, $\varphi (e)$ is the degenerate edge $\operatorname{id}_{X}$ of $\operatorname{\mathcal{C}}$. Since $\varphi $ is a right fibration (Proposition 4.3.6.1), it follows that $e$ is an isomorphism in $\operatorname{\mathcal{C}}_{/Z}$ (Proposition 4.4.2.11). Applying Proposition 4.4.5.8, we deduce that the lifting problem
\[ \xymatrix@C =50pt@R=50pt{ (\Delta ^1 \times \Lambda ^{n}_{n} ) {\coprod }_{ ( \{ 0\} \times \Lambda ^{n}_{n} )} ( \{ 0\} \times \operatorname{\partial \Delta }^{n} ) \ar [r]^-{(h'_{+}, f')} \ar [d] & \operatorname{\mathcal{C}}_{/Z} \ar [d]^{\varphi } \\ \Delta ^1 \times \operatorname{\partial \Delta }^{n} \ar [r]^-{h} \ar@ {-->}[ur]^{h'} & \operatorname{\mathcal{C}}} \]
admits a solution. The morphism $h'$ is then a natural transformation from $f'$ to a morphism $g': \operatorname{\partial \Delta }^{n} \rightarrow \operatorname{\mathcal{C}}_{/Z}$, which we can identify with a map $\overline{g}: ( \operatorname{\partial \Delta }^{n} )^{\triangleright } \rightarrow \operatorname{\mathcal{C}}$ satisfying $\overline{g}|_{ \operatorname{\partial \Delta }^{n} } = g$.
$\square$