A lattice-counting conjecture for paired quaternionic matrices

Let $g\in \operatorname{SL}_{2}(\mathbb{R})$. For parameters $\ell$, $\delta$, $L$, and $\heartsuit$, consider pairs $\gamma_1,\gamma_2\in R(\ell;g)$ lying in $\Omega^\star(\delta,L)\cup\Psi^\star(\delta,L)$, with equal determinants and satisfying

$$|\diamondsuit(\gamma_1)-\diamondsuit(\gamma_2)|\leq \heartsuit L^4.$$

Here $A\prec B$ means $A\ll_{\epsilon}(d_BN(1+L))^{\epsilon}B$ for every sufficiently small positive $\epsilon$. Lattice-counting conjecture. One has

$$\sum_{\substack{\gamma_1,\gamma_2 \in R(\ell;g)\\ \gamma_1,\gamma_2 \in \Omega^\star(\delta,L) \cup \Psi^\star(\delta,L) \\ \det(\gamma_1)=\det(\gamma_2) \\ |\diamondsuit(\gamma_1)-\diamondsuit(\gamma_2)| \leq \heartsuit L^4}} 1 \prec \ell L^2 \Biggl(1 +\frac{\ell^2}{d_BN} \delta^{\frac{1}{2}}L^2 + \frac{\ell^4}{(d_BN)^2} \min\{\heartsuit, \delta\} L^4 + \ell H(g)^2 + \ell^{\frac{3}{2}} H(g)^2 \delta^{\frac{1}{4}} L+\ell^2 H(g)^2 \delta^{\frac{1}{2}}L^2 \Biggr).$$

The terms involving $H(g)$ are omitted when $B$ is non-split, equivalently when $d_B>1$. This estimate is one of the counting inputs for the paper's sup-norm and fourth-moment results; its status is not resolved in the supplied context.