Thurston's genericity conjecture for pseudo-Anosov mapping classes

Let $\Sigma$ be a surface, let $\operatorname{MCG}(\Sigma)$ denote its mapping class group, and let $S$ be a finite generating set for $\operatorname{MCG}(\Sigma)$. Write $d_S$ for the corresponding word metric and let $e$ be the identity. Thurston's genericity conjecture. The proportion of pseudo-Anosov mapping classes in the word-metric ball of radius $k$ tends to $1$:

$$\frac{\#\{g \in \operatorname{MCG}(\Sigma): g \text{ is pseudo-Anosov and } d_S(e,g) \leq k\}}{\# \{g \in \operatorname{MCG}(\Sigma): d_S(e,g) \leq k\}} \longrightarrow 1.$$

This conjecture predicts that pseudo-Anosov mapping classes are exponentially generic, and is a long-standing conjecture attributed to Thurston and collected by Farb. Its status is not resolved in the supplied source.