Nonvanishing random entries in the character table of the symmetric group

Let $\lambda,\mu\vdash n$ be independently chosen random partitions of $n$, and let $\chi^\lambda[\mu]$ denote the value of the irreducible character indexed by $\lambda$ on the conjugacy class indexed by $\mu$. Random-character-table conjecture. Then

$$\chi^\lambda[\mu]\ne0$$

with high probability as $n\to\infty$. This is posed as a natural question about random entries of the character table of $S_n$; the source gives no resolution and leaves it open.