Joint generalized-Fibonacci form of pre-moments

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.

Joint pre-moment conjecture. The $k$th pre-moment $m_k(d,n)$ of $X_{d,n}$ is of the form

$$A(n,d)N_d(n)+B(n,d)N_d(n+1),$$

where $A$ and $B$ are degree $2k$ polynomials in $n$ whose coefficients are rational functions in $d$.

This gives a single ansatz for the pre-moments when both parameters vary, extending the fixed-$d$ form. The source reports computational evidence and explicitly notes the computational cost, but gives no proof or resolution.