Zeta-polynomial conjecture for m-divisible odd noncrossing partitions

For $m\geq 1$, let $C=(x_1,x_2,\ldots,x_m)$ be an $m$-multichain in $\mathcal{O\!N\C}_{2n+1}$, with extended delta sequence $\delta_o(C)=(d_0;d_1,d_2,\ldots,d_m)$ defined by $x_0=e$, $x_{m+1}=c$, and $d_i=x_i^{-1}x_{i+1}$. Define $\mathcal{O\!N\C}_{2n+1}^{(m)}$ by $C\leq C'$ if and only if $d_i\geq_{\mathbf{3}}d_i'$ for $i\in[m]$. Zeta-polynomial conjecture. For $n,m\geq 1$, the zeta polynomial of $\mathcal{O\!N\C}_{2n+1}^{(m)}$ is

$$\mathcal{Z}\bigl(\mathcal{O\!N\C}_{2n+1}^{(m)},q\bigr)=\frac{m(q-1)+1}{(2m(q-1)+1)n+m(q-1)+1}\binom{(2m(q-1)+1)n+m(q-1)+1}{n}.$$

This conjecture gives an explicit enumeration of multichains in the $m$-divisible odd noncrossing-partition poset; the source provides no resolution or further evidence.