The pseudo-Anosov prevalence conjecture for simple braids

Let $\Br_n$ be the braid group on $n$ strands, identified with the mapping class group of a disk with $n$ marked points relative to its boundary. A braid is pseudo-Anosov if its mapping class has pseudo-Anosov Nielsen–Thurston type. Pseudo-Anosov prevalence conjecture. The proportion of simple braids on $n$ strands that are pseudo-Anosov tends to $100\%$ as $n$ tends to infinity.

This would sharpen the result that the proportion of simple braids with positive topological entropy tends to $100\%$, and would be complementary to results of Caruso and Wiest for braids in Cayley balls. Since pseudo-Anosov type is stronger than positive entropy, the proposed statement is left as future work.