The geometric P=W conjecture for Betti and Dolbeault moduli spaces

Let $N_{Dol}^*$ be the complement of a large Hitchin-map sublevel set in $\mathcal{M}_{Dol}$, let $N_B^*$ be a punctured tubular neighborhood of the boundary divisor in a log compactification of $\mathcal{M}_B$, and let $\overline{h}:N_{Dol}^*\to S^{d-1}$ and $\alpha:N_B^*\to\mathbb{D}\partial\mathcal{M}_B$ be the corresponding fibrations up to homotopy. Here $d=\dim_{\mathbb{C}}\mathcal{M}_B$. Geometric P=W conjecture. There exists a homotopy commutative square

$$\begin{CD} N_{Dol}^* @>{\simeq\ \phi}>> N_B^*\\ @V{\overline{h}}VV @VV{\alpha}V\\ S^{d-1} @>{\simeq}>> \mathbb{D}\partial\mathcal{M}_B \end{CD}$$

where the top arrow is a homotopy equivalence induced by the non-abelian Hodge correspondence and the bottom arrow is the homotopy equivalence in the homotopy type conjecture. This conjecture compares the Hitchin fibration at infinity with the boundary fibration on the Betti side and remains open in general.