Discreteness conjecture for commensurators of subgroups of lattices in $\mathrm{PSL}_2(\mathbb{R})$

Let $H<\mathrm{PSL}_2(\mathbb{Z})$ be a finite index subgroup with $b_1(H)\geq 1$. Let $\Phi(H)$ denote the subgroup introduced in the paper, and let $K$ be an infinite index normal subgroup of a lattice $\Gamma<\mathrm{PSL}_2(\mathbb{R})$ such that $|K|=\infty$. For a subgroup $L<\mathrm{PSL}_2(\mathbb{R})$, write $\operatorname{Comm}_{\mathrm{PSL}_2(\mathbb{R})}(L)$ for its commensurator.

Commensurator discreteness conjecture. The group $\operatorname{Comm}_{\mathrm{PSL}_2(\mathbb{R})}(\Phi(H))$ is discrete provided that $H\neq\Phi(H)$. More generally,

$$\operatorname{Comm}_{\mathrm{PSL}_2(\mathbb{R})}(K)\text{ is discrete}.$$

These statements would extend the paper's discreteness theorem beyond the hypothesis that $H$ is contained as a normal subgroup of a principal congruence subgroup. The source presents them as expected generalizations, and does not give evidence that they have been resolved.