Multinomial power-sum congruence conjecture

For integers $m,n\geq 1$ and $r\geq 0$, define the multinomial power sums

$$M_{m,n}^{(r)}=\sum_{k_1+\cdots+k_m=n}{n\choose k_1,\ldots,k_m}^r,$$

where $k_1,\ldots,k_m\geq 0$ and $k_1+\cdots+k_m=n$. The multinomial congruence conjecture. Let $m,n\geq 1$ and $r\geq 0$ be integers. Then

$$\sum_{k=0}^{n-1}(-1)^{rk}((m+1)k+m)M_{m,k}^{(r)}\equiv 0\pmod {mn}.$$

This further generalizes the proposed Franel-number congruence by replacing binomial power sums with multinomial power sums. Its status is not resolved in the supplied text.