Generalization of the Smith normal form theorem for the action of k partial p_k

The map $k\frac{\partial}{\partial p_k}p_k$ acts on homogeneous symmetric functions of degree $n$. With respect to an integral basis, let $M_k^{(n)}$ denote its matrix, and let $p(n)$ be the number of partitions of $n$. Let $\mathcal{M}$ be the multiset of all numbers $m_k(\lambda)$ for $\lambda\vdash n$. Generalized Smith normal form conjecture. There exist $P(x),Q(x)\in\mathrm{SL}(p(n),\mathbb{Z}[x])$ such that

$$P(x)(M_k^{(n)}+xI_{p(n)})Q(x)=\operatorname{diag}(f_1(x),\dotsc,f_{p(n)}(x)),$$

where the diagonal entries are determined successively from the distinct elements of $\mathcal{M}$: $f_{p(n)}(x)$ is a product of factors $x+k(a_i+1)$, with the $a_i$ ranging over the distinct elements of $\mathcal{M}$; $f_{p(n)-1}(x)$ is a product of factors $x+k(b_i+1)$, with the $b_i$ ranging over the remaining distinct elements of $\mathcal{M}$; and so on, until all elements of $\mathcal{M}$ have been exhausted, with the remaining diagonal entries equal to the empty product $1$. This conjectures that the theorem proved in the paper extends from its original action to the action $k\frac{\partial}{\partial p_k}p_k$ for every $k\geq1$.