The incidence-complex and Hitchin-base sphere correspondence

Let $M_B$ be the character variety with a compactification whose boundary is a normal-crossings divisor, and let $M_{Dol}$ be the Dolbeault moduli space with Hitchin base of complex dimension $N$. Write the boundary as

$$\overline{M}_{B}-M_{B}=\bigcup_iD_i,$$

and let $|\operatorname{Step}(M_B)|$ be the realization of the simplicial incidence complex having one $n$-simplex for each connected component of $D_{i_0}\cap\cdots\cap D_{i_n}$. Let $N_B$ and $N_{Dol}$ be small neighborhoods of the divisors at infinity intersected with $M_B$ and $M_{Dol}$, respectively. The incidence-complex and Hitchin-base sphere conjecture. There is a homotopy-commutative diagram

$$\begin{array}{ccc} N_{Dol} & \stackrel{\sim}{\longrightarrow} & N_B \\ \downarrow & & \downarrow \\ S^{2N-1} & \stackrel{\sim}{\longrightarrow} & |\operatorname{Step}(M_B)|. \end{array}$$

The incidence complex is independent of the chosen compactification up to homotopy, and the conjecture identifies it with the sphere at infinity of the Hitchin base through the nonabelian Hodge correspondence. The source first motivates this relationship in the rank-two four-punctured-sphere example, but gives no resolution of the conjecture.