Two-scale asymptotic normality for distinct-part core-partition sizes

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 $\sigma_{d,n}$. For each fixed $n$, consider the standardized variable $(X_{d,n}-\mu_{d,n})/\sigma_{d,n}$ as $d$ tends to infinity.

Two-scale normality conjecture. For each fixed $n$, the distribution of $X_{d,n}$ is not asymptotically normal; in fact, $(X_{d,n}-\mu_{d,n})/\sigma_{d,n}$ tends to some abnormal distribution $X_n$ as $d\to\infty$. However, $X_n$ is asymptotically normal: $(X_n-\mu)/\sigma$ tends to the standard normal distribution as $n\to\infty$.

The conjecture distinguishes the limits in $d$ and $n$: the first limit is non-normal for fixed $n$, while the resulting family is expected to become normal as $n$ grows. The source supports this with computed standardized moments but gives no proof or resolution.