Hypergeometric formula for matching polynomials of complete multipartite graphs

Let $K_{(n)}$ be the complete $k$-partite graph with $N=n_1+\dots+n_k$ vertices, whose $k$ color classes have sizes $n_1,\dots,n_k$. For odd $r$, let $M_r(K_{(n)})$ denote its matching polynomial, and let $\Delta(r+1; a)$ denote the parameter list associated with the step $r+1$. The complete multipartite matching-polynomial conjecture. For odd $r$, the matching polynomials are given by

$$M_r(K_{(n)}) = x^N \,{}_{r+1}F_{0} \left[\begin{matrix} \Delta(r+1; -n_1)\ldots \Delta(r+1; -n_k) \\ - \end{matrix}; \frac{-(r+1)^{r+1}}{(2x)^{r+1}}\right].$$

This is suggested by the established formulas for complete graphs and complete bipartite graphs, but the source does not provide a proof or resolution for the complete $k$-partite case.