Equality of limiting zero proportions in symmetric-group character tables

For each $n$, let $S_n$ be the symmetric group, and let $z_I(S_n)$, $z_{I\!I}(S_n)$, and $z(S_n)$ denote respectively the proportions of character-table zeros of type I, of type II, and of all zeros in the character table of $S_n$. Equality conjecture. The three limiting proportions exist and are equal:

$$\lim_{n\to\infty}z_I(S_n)=\lim_{n\to\infty}z_{I\!I}(S_n)=\lim_{n\to\infty}z(S_n).$$

The equality is proposed on the basis of the reported Monte Carlo data, and no proof or resolution is given.