Object-space equivalence conjecture for iterated enriched categories

Let $\mathcal{C}$ be an $(\infty,n)$-category, and let $\mathrm{O}_{n-1}$ and $\Theta_n$ be the indexing shapes used to define the iterated constructions $\dot{\mathcal{T}}^n(\mathcal{C})$ and $\dot{\mathcal{L}}^n(\mathcal{C})$. Object-space equivalence conjecture. The natural maps

$$\mathrm{Fun}(\mathrm{O}_{n-1},\mathcal{C})^\simeq\longrightarrow\dot{\mathcal{T}}^n(\mathcal{C})^\simeq$$

and

$$\mathrm{Fun}(\Theta_n,\mathcal{C})^\simeq\longrightarrow\dot{\mathcal{L}}^n(\mathcal{C})^\simeq$$

are equivalences of spaces. The preceding lemma proves surjectivity on $\pi_0$ for the first map; the conjecture asserts the stronger equivalence statement for both maps.