Folklore conjecture on density of pseudo-Anosovs in the mapping class group

Fix a finite generating set for ${\rm Mod}_g$, let $B(r)$ be the ball of radius $r$, and define the density of $S\subset {\rm Mod}_g$ by

$$d(S)=\lim_{r\to\infty}\frac{\#(B(r)\cap S)}{\#B(r)}.$$

Let ${\cal P}$ be the set of pseudo-Anosov elements of ${\rm Mod}_g$. Density conjecture for pseudo-Anosovs.

$$d({\cal P})=1.$$

Random-walk and other probabilistic results show pseudo-Anosovs are prevalent in related senses, but the stated density conjecture does not follow directly from those results.