Random character values on self-conjugate principal hooks

Let $\lambda\vdash n$ be a uniformly random partition of $n$, and let $\mu$ be a uniformly random self-conjugate partition of $n$. Write $\widehat\mu$ for the principal-hook partition of $\mu$. Random self-conjugate-character conjecture. Then

$$\chi^\lambda[\widehat\mu]=0$$

with high probability as $n\to\infty$. The claim is motivated heuristically by rim-hook and limit-shape considerations, while the source gives only partial evidence and leaves the conjecture open.