Asymptotic normality conjecture for distinct-part core-partition sizes

For each positive integer $s$, let $X_s$ be the size of a uniformly chosen $(s,s+1)$-core partition with distinct parts. Let $\mathbb{E}[X_s]$ and $\operatorname{Var}(X_s)$ denote its expectation and variance. Asymptotic normality conjecture. The distribution of

$$\frac{X_s-\mathbb{E}[X_s]}{\sqrt{\operatorname{Var}(X_s)}}$$

converges to the standard normal distribution as $s\to\infty$. The conjecture is motivated by the agreement of the first 16 limiting standardized central moments with those of the normal distribution; the source presents this as evidence rather than a proof.