Sparse large sieve conjecture for function fields

Let $\boldsymbol{F}_q[t]$ be the polynomial ring over the finite field with $q$ elements. Let $\boldsymbol{S}$ be a set of nonzero monic polynomials in $\boldsymbol{F}_q[t]$ of degree at most $Q$, where $Q$ and $N$ are positive integers, and let $(a_g)$ be arbitrary complex numbers indexed by polynomials $g\in\boldsymbol{F}_q[t]$ with $\deg g\leq N$. For $\varepsilon>0$, the conjecture asserts

$$\sum_{f\in S}\sum_{\substack{r\bmod f\\(r,f)=1}}\left|\sum_{\substack{g\in\boldsymbol{F}_q[t]\deg g\leq N}}a_g e\left(g\cdot\frac{r}{f}\right)\right|^2 \ll_{q,\varepsilon}q^{\varepsilon(Q+N)}\left(q^N+\left(\sharp S\right)q^Q\right)\sum_{\substack{g\in\boldsymbol{F}_q[t]\deg g\leq N}}|a_g|^2.$$

This is a plausible weakening of a previously claimed sparse large sieve bound, whose general validity was disproved by a counterexample; whether this weakened estimate holds for arbitrary sets $S$, or requires additional conditions, is unknown.}