Let $\mathrm{Diff}_n$, $\mathrm{PL}_n$, $\mathrm{Top}_n$ denote the $\infty$-categories of $n$-manifolds which are respectively smooth/PL/topological, and open embeddings (for instance by taking the homotopy coherent nerve of the sing of the corresponding topological categories).

Let $\operatorname{BTop}(n)$ and $ \mathrm{B}O(n)$ be the classifying spaces of topological respectively orthogonal $\mathbb{R}^n$-bundles. Similarly let $\operatorname{BPL}(n)$ denote the classifying space of $\mathrm{PL}$-bundles of rank $n$ (it's not quite the classifying space of a topological group but is actually the simplicial set classifying $\mathrm{PL}$ bundles over polyhedra). Let $\mathcal{S}_{/X}$ denote the slice category of the $\infty$-category of spaces over a fixed space $X$. There's a canonical commutative diagram of $\infty$-categories (for the middle row in this diagram you have to work a bit but I'm pretty sure this is true): $\require{AMScd}$

\begin{CD} \mathrm{Diff}_n @>>> \mathrm{PL}_n @>>> \mathrm{Top}_n\\ @VVV @VVV @VVV \\ \mathcal{S}_{/\mathrm{B}O(n)} @>>> S_{ /\mathrm{B}\mathrm{PL}(n)} @>>> \mathcal{S}_{/\mathrm{BTop}(n)} \end{CD}

Question:(for $n \ne 4$) Are all the squares in this diagram pullback squares? If so where can I find this or at least the relevant pieces of the argument?