The exponential-formula conjecture for primitive ideals in quantum matrices

Let $m\ge 1$ be a positive integer, and let $P(m,n)$ denote the number of primitive $\mathcal{H}$-prime ideals in ${\mathcal O}_q(M_{m,n})$. Then there are rational constants $c_{m+1},c_m,\ldots,c_{2-m}$.

Exponential enumeration conjecture. For every positive integer $n$,

$$P(m,n)=\sum_{j=2-m}^{m+1}c_jj^n,$$

and

$$c_{m+1}=\frac{1\cdot3\cdot5\cdots(2m-1)}{2^m}.$$

This is proposed as a generalization of the known formulas for one- and two-row quantum matrices; if true, it would imply the asymptotic conjecture about the proportion of primitive $\mathcal{H}$-primes. Its resolution is not indicated in the source.