Let $\mathcal{C}$ be an $\infty$-category and consider the outer horn inclusion $\Lambda[3]_3 \subset \Delta[3]$. Given a diagram $f:\Lambda[3]_3 \to \mathcal{C}$, if the image of $\Delta^{\{2,3\}}$ is an equivalence in $\mathcal{C}$ then there exists a lift to $\Delta[3]$, unique up to a contractible space of choice. Suppose now that we only have $\mathrm{Map}_{\mathcal{C}}(f(\Delta^{\{0\}}),f(\Delta^{\{2\}}))\to \mathrm{Map}_{\mathcal{C}}(f(\Delta^{\{0\}}),f(\Delta^{\{3\}}))$ an equivalence in $\mathcal{S}$, where we use the mapping space bifunctor $\mathcal{C}^{op}\times \mathcal{C}\to \mathcal{S}$ to the category $\mathcal{S}$ of spaces. Does the same result hold ?
(In the $1$-categorical setting it is certainly true ; we want to show that the image of the two morphisms $0\to 2$ and $0 \to 1 \to 2$ coincide in $\mathcal{C}$ ; but they coincide after composition with the image of $2\to3$, so we are done.)
What if $\mathcal{C}$ is actually a category of functors $\mathrm{Fun}(K,\mathcal{D})$ from a simplicial set $K$ to an $\infty$-category $\mathcal{D}$, and we only have $\mathrm{Map}_{\mathcal{D}}(f(\Delta^{\{0\}})(k),f(\Delta^{\{2\}})(k))\to \mathrm{Map}_{\mathcal{D}}(f(\Delta^{\{0\}})(k),f(\Delta^{\{3\}})(k))$ an equivalence in $\mathcal{S}$ for every vertex $k\in K$ ?