Asymptotic representation-growth conjecture for Fuchsian groups

Let $\Gamma$ be the Fuchsian group from the setup, with orbifold Euler characteristic

$$\chi(\Gamma)=2-2g-\sum_{i=1}^r\left(1-\frac{1}{a_i}\right)<0.$$

Let $A$ be the least common multiple of $a_1,\ldots,a_r$, with $A=1$ if $r=0$, and let $q$ be a prime power relatively prime to $A$. Write $\operatorname{GL}_n(q)$ for the general linear group over $\mathbb{F}_q$.

Representation-growth conjecture. (a) There exists a $2A$-periodic sequence $c_{q,1},c_{q,2},\ldots$ of positive numbers such that

$$|\operatorname{Hom}(\Gamma,\operatorname{GL}_n(q))|\sim c_{q,n}q^{(1-\chi(\Gamma))n^2}.$$

(b) There exist a $2$-dimensional array $e_{\Gamma,q,n}$ of rational numbers and a $2$-dimensional array $f_{\Gamma,q,n}$ of half-integral weight meromorphic modular forms, periodic in both $q$ and $n$, such that

$$c_{q,n}=(q-1)q^{e_{\Gamma,q,n}}f_{\Gamma,q,n}\left(\frac{i\log q}{2\pi}\right).$$

Moreover, $f_{\Gamma,q,n}$ is holomorphic on the upper half plane and has integer Fourier coefficients at $i\infty$.

The conjecture predicts both the leading asymptotic growth of homomorphisms from $\Gamma$ to finite general linear groups and a modular-form description of the periodic leading constants. The supplied text gives no resolution evidence, so the conjecture is recorded as open.