Argyres-Douglas modular representation conjecture for the DAHA module

Let $\scD^{(1)}_{\ell}$ be the $\ell$-dimensional module singled out among the four modules in the shortening regime, with basis indexed by $0\leq j\leq \ell-1$. Let $\widetilde S_{jj'}$ be the transformed $S$-matrix and let $b_\ell$ and $g_j$ be the normalization factors appearing in the formulas. Argyres-Douglas modular representation conjecture. The space $\scD^{(1)}_{\ell}$ is an $\ell$-dimensional $\mathrm{PSL}(2,\mathbb{Z})$ representation with the displayed modular $S$ and $T$ matrices; this representation comes from a modular tensor category associated to the Argyres-Douglas theory of type $(A_1,A_{2(\ell-1)})$, and the matrices coincide with those of the $(2,2\ell+1)$ Virasoro minimal model at $q=e^{-2\pi i/(2\ell+1)}$. The claim connects the DAHA module to both an Argyres-Douglas modular tensor category and a Virasoro minimal model, but no resolution status is supplied.