The asymptotic proportion conjecture for primitive ideals in quantum matrices

Let $m$ be a fixed positive integer. Consider the $\mathcal{H}$-prime ideals of the quantized coordinate ring ${\mathcal O}_q(M_{m,n})$.

Primitive-proportion conjecture. As $n\rightarrow\infty$, the proportion of $\mathcal{H}$-primes that are primitive tends to

$$\frac{{2m \choose m}}{4^m}.$$

The source states that this conjecture would follow from the exponential formula conjecture for $P(m,n)$. No resolution is indicated.