Asymptotic normality for fixed $d$

Let $X_{d,n}$ be the size of a uniformly random $(n,dn-1)$-core partition with distinct parts, with mean $\mu_{d,n}$ and standard deviation $\sigma_{d,n}$. Fix $d$ and let $n$ tend to infinity.

Fixed-$d$ normality conjecture. For fixed $d$, the distribution of $X_{d,n}$ is asymptotically normal. That is,

$$\frac{X_{d,n}-\mu_{d,n}}{\sigma_{d,n}}$$

approaches the standard normal distribution as $n\to\infty$.

The conjecture is motivated by the observed limiting standardized moments $0,1,0,3,0,15,\dots$. It is presented as an experimentally supported conjecture, with no proof or resolution stated.