Generalized-Fibonacci form of pre-moments for fixed $d$

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 $m_k(d,n)$ denote its $k$th pre-moment, and let $N_d(n)$ denote the generalized Fibonacci sequence used in the source.

Generalized-Fibonacci pre-moment conjecture. For each $d$, the $k$th pre-moment $m_k(d,n)$ of $X_{d,n}$ is of the form

$$a(n)N_d(n)+b(n)N_d(n+1),$$

where $a$ and $b$ are polynomials in $n$.

This conjecture extends the corresponding fixed-parameter formulas known in the source's earlier work. The source says that experimental evidence verifies the claim, but supplies no proof or database resolution.