Classification conjecture for arithmetic discrete groups from rigid DAHA modules

Let $\mathcal{H}$ and $\mathcal{H}^{\checkmark}$ denote the two DAHA-type algebras considered in the paper, and let $\mathcal{B}_q$ be the associated braid-group representation. Consider irreducible $\mathcal{H}$-modules and irreducible $\mathcal{H}^{\checkmark}$-modules on which $X$ and $Y$ act semisimply. Identify representations up to isomorphism and up to changing the signs of the images of $T$, $Y$, and $X$. Classification conjecture. The arithmetic discrete non-finite groups $\operatorname{Image}(\mathcal{B}_q)$ can be only those appearing in the lists corresponding to

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. This asserts that the displayed lists exhaust the arithmetic discrete non-finite groups arising under these hypotheses; the source provides no resolution status for the assertion.