Refined Chern-Simons representation conjecture for the spherical DAHA module

Let $q$ and $t$ satisfy the shortening condition defining the $(k+1)$-dimensional module $\repV_{k+1}$ of the spherical double affine Hecke algebra, with basis indexed by $0\leq j\leq k$. Let $P_j$ be the corresponding Macdonald polynomials, and let $g_j$ and $a_k$ denote the normalization factors in the stated formulas. Refined Chern-Simons representation conjecture. The space $\repV_{k+1}$ is a $(k+1)$-dimensional representation of $\mathrm{PSL}(2,\mathbb{Z})$, with modular matrices given by the displayed formulas for $T_{jj'}$ and $S_{jj'}$; these matrices provide the representation for refined Chern-Simons theory. This identifies the DAHA construction with the modular representation expected from refined Chern-Simons theory, but the source supplies no resolution evidence beyond the asserted result.