The asymptotic-dimension conjecture for the mapping class group

Let $S$ be the finite-type surface under consideration, of genus $g$, and let ${\rm Mod}(S)$ denote its mapping class group. Write ${\rm asdim}({\rm Mod}(S))$ for its asymptotic dimension and ${\rm vd}({\rm Mod}(S))$ for its virtual cohomological dimension.

Asymptotic-dimension conjecture.

$${\rm asdim}({\rm Mod}(S))={\rm vd}({\rm Mod}(S))=4g-5.$$

This conjecture, attributed to Bestvina and Bromberg, predicts that the asymptotic dimension of the mapping class group equals its virtual cohomological dimension, with both given by $4g-5$. The surrounding results provide support for the conjecture, but the supplied text does not state that it has been resolved.