Thanatipanonda's generalized binomial-power expansion conjecture

Fix integers $d$, $k\geq 0$, and $r\geq 1$. The coefficients $a^{(r)}_{d,k,j}$ are intended to expand powers of $\binom{n+dk}{k}$ in the family $\binom{n+dj}{j}$. Generalized expansion conjecture. For all such $d,k,r$, there exist integers $a^{(r)}_{d,k,j}$ such that

$$\binom{n+dk}{k}^r=\sum_j a^{(r)}_{d,k,j}\binom{n+dj}{j}\qquad\text{for all }n=0,1,2,\ldots.$$

They can be initialized by $a^{(1)}_{d,k,k}=1$ and $a^{(1)}_{d,k,j}=0$ for $j\ne k$, and recursively defined by

$$a^{(r+1)}_{d,k,j}=\sum_i S_d(k,j,i)a^{(r)}_{d,k,i},$$

where the $S_d(k,j,i)$ are integers independent of $r$, for all $d,k,j,i$. The statement extends cases proved for $d=0,1$ to all non-negative integers $d$; the source gives no later resolution.