Genericity conjecture for pseudo-Anosov mapping classes in metric balls

Let $\Sigma$ be a complete hyperbolic surface of finite type that is not a three-punctured sphere, and let $\operatorname{Mod}(\Sigma)$ be its mapping class group. Let $S$ be a specific or arbitrary finite generating set of $\operatorname{Mod}(\Sigma)$, and let $B_S(R)$ be the collection of mapping classes whose word norm with respect to $S$ is at most $R$. Genericity conjecture.

$$\lim_{R \rightarrow \infty} \frac{\# \{g \in B_S(R): g\text{ is pseudo-Anosov}\}}{\#B_S(R)}=1.$$

This conjecture expresses the belief that pseudo-Anosov mapping classes are generic in metric balls of the mapping class group, independently of the chosen finite generating set. It is presented as one of two metric interpretations of the expectation that most mapping classes are pseudo-Anosov; the supplied text does not state whether it has been resolved.