Random character values on staircase principal hooks

Let $P_n$ be the set of partitions of $n$, and let $\widehat\rho_k$ denote the principal-hook partition of the staircase partition $\rho_k=(k,k-1,\ldots,1)$, where $n=k(k+1)/2$. Choose $\lambda\in P_n$ uniformly at random. Random-staircase-character conjecture. Then

$$\chi^\lambda[\widehat\rho_k]\ne0$$

with high probability as $n\to\infty$. This conjecture concerns the typical nonvanishing of symmetric-group character values; the source gives no proof and leaves it open.