Polynomiality and degree of pre-moments in the parameter $d$

Let $X_{d,n}$ be the size of a uniformly random $(n,dn-1)$-core partition with distinct parts. For a natural number $k$, let $G_{d,n}(q)$ be the generating function of $X_{d,n}$ and define its $k$th pre-moment by

$$m_k(d,n):=\left[\left(q\frac{d}{dq}\right)^kG_{d,n}(q)\right]_{q=1}.$$

Polynomial pre-moment conjecture. For each $n$, the $k$th pre-moment $m_k(d,n)$ of $X_{d,n}$ is a polynomial in $d$. Further, the degree of this polynomial is $2k+\lfloor n/2\rfloor$.

Such polynomial formulas would give explicit control of all moments when $n$ is fixed and $d$ varies. The claim is presented as an experimental observation, with no proof or resolution supplied in the source.