Asymptotic equality of mod-$\ell$ and $\ell$-adic rank-2 representation counts

Let $X$ be a smooth projective curve over ${\mathbb F}_q$, let $\ell$ be a prime different from the characteristic of ${\mathbb F}_q$, and let $T(X,2,q^n,\ell)$ and $\overline{T(X,2,q^n,\ell)}$ denote respectively the cardinalities of the twisting-equivalence classes of two-dimensional $\ell$-adic and mod-$\ell$ representations that remain irreducible on restriction to the geometric fundamental group. Likewise, let $T'(X,2,q^n,\ell)$ and $\overline{T'(X,2,q^n,\ell)}$ denote the corresponding cardinalities without the requirement that the representations be fixed by Frobenius. Asymptotic lifting conjecture. The ratios satisfy

$$\lim_{n\rightarrow\infty}\frac{\overline{T(X,2,q^n,\ell)}}{T(X,2,q^n,\ell)}=1$$

as $n$ runs through all positive integers prime to $\ell$, and

$$\lim_{n\rightarrow\infty}\frac{\overline{T'(X,2,q^n,\ell)}}{T'(X,2,q^n,\ell)}=1$$

as $n$ runs through all positive integers. The conjecture is motivated by the expectation that there are few congruences modulo $\ell$ between representations as $n$ varies; de Jong's lifting result gives the opposite inequality between the mod-$\ell$ and $\ell$-adic counts, but does not establish these limiting equalities.