The twisted Linnik–Selberg conjecture over function fields

Let $K=\mathbb{F}_q(t)$, $K_{\infty}=\mathbb{F}_q(!(1/t)!)$, and let $\psi$ be the nontrivial additive character on $K_{\infty}$ that is trivial on $\mathbb{F}_q[t]$. For nonzero $r\in\mathbb{F}_q[t]$, define $\psi_r(x)=\psi(x/r)$ on $\mathbb{F}_q[t]/(r)$, and for $m,n$ in the subring of elements with denominator relatively prime to $r$, define

$$\operatorname{Kl}_r(m,n)=\sum_{x\in(\mathbb{F}_q[t]/(r))^*}\psi_r(mx+n\overline{x}).$$

Define $\operatorname{Kl}_{\infty}(\psi,\alpha)$ by the infinite-place integral given in the source. Twisted Linnik–Selberg conjecture over function fields. For nonzero $g\in\mathbb{F}_q[t]$, $\delta$ relatively prime to $g$, integer $T\ge0$, $\alpha\in\mathbb{F}_q[t]$, and nonzero $a,b\in\mathbb{F}_q[t,g^{-1}]$, the two sums displayed in the source are each bounded by $|gab|_{\infty}^{\varepsilon}\widehat{T}^{1+\varepsilon}$, with implied constant depending on $\varepsilon,\delta$:

$$\left|\sum_{\substack{|r|=\widehat{T}\\ (g,r)=1,\ \delta\mid r}}\psi_{g^2}(\alpha r^{-1})\operatorname{Kl}_r(a,b)\right|\ll_{\varepsilon,\delta}|gab|_{\infty}^{\varepsilon}\widehat{T}^{1+\varepsilon}.$$

Furthermore,

$$\left|\sum_{\substack{|r|=\widehat{T}\\ (g,r)=1,\ \delta\mid r}}\psi_{g^2}(\alpha r^{-1})\operatorname{Kl}_r(a,b)\operatorname{Kl}_{\infty}(\psi,ab/r^2)\right|\ll_{\varepsilon,\delta}|gab|_{\infty}^{\varepsilon}\widehat{T}^{1+\varepsilon}.$$

The conjecture asks for an additional square-root cancellation beyond Weil's estimate for Kloosterman sums and is presented as a twisted function-field analogue of the Linnik–Selberg conjecture.