Hausel–Rodriguez-Villegas boundary-complex conjecture for character varieties

Let $\mathcal{R}_{g,\boldsymbol{\mu}}$ be the character variety, let $\overline{\mathcal{R}_{g,\boldsymbol{\mu}}}$ be the compactification constructed from the representation variety, and let $D^{{\it B}}_{g,\boldsymbol{\mu}}$ be its boundary divisor. Let $N^{{\it B}}_{g,\boldsymbol{\mu}}$ be a small punctured neighborhood of this divisor, and let $\Delta(D^{{\it B}}_{g,\boldsymbol{\mu}})$ be its boundary complex. Let $\mathcal{M}_{g,\boldsymbol{\mu}}$ be the moduli space of parabolic Higgs bundles, of dimension $d_{g,\boldsymbol{\mu}}$, and let $N^{{\it Dol}}_{g,\boldsymbol{\mu}}$ be a small punctured neighborhood of the divisor at infinity in its canonical orbifold compactification. The Hitchin fibration gives a map $N^{{\it Dol}}_{g,\boldsymbol{\mu}}\to S^{d_{g,\boldsymbol{\mu}}-1}$. Boundary-complex conjecture. There exists a homotopy-commutative diagram

$$\begin{CD} N^{{\it Dol}}_{g,\boldsymbol{\mu}} @>{\cong}>> N^{{\it B}}_{g,\boldsymbol{\mu}} \\ @VVV @VVV \\ S^{d_{g,\boldsymbol{\mu}} -1} @>{\cong}>>\Delta(D^{{\it B}}_{g,\boldsymbol{\mu}}). \end{CD}$$

In particular, there exists a nonsingular compactification of $\mathcal{R}_{g,\boldsymbol{\mu}}$ whose boundary complex is a simplicial decomposition of $S^{d_{g,\boldsymbol{\mu}}-1}$. The conjecture relates the topology of character varieties, their compactifications, and the Hitchin fibration; the supplied text gives no general resolution.