Berest–Samuelson's DAHA action conjecture for knot skein modules

Let $K\subset S^3$ be a knot, let $\sk_q(S^3\setminus K)$ be the Kauffman bracket skein module of its complement, and let $\mathrm{SH}_{q,t_1,t_2,1,1}$ denote the spherical double affine Hecke algebra with parameters $q,t_1,t_2,1,1$. Berest–Samuelson's DAHA action conjecture. The algebra $\mathrm{SH}_{q,t_1,t_2,1,1}$ acts canonically on $\sk_q(S^3\setminus K)$. This conjecture was proposed on the basis of explicit computations and its $q=-1$ specialization gives the conjectural regular deformation of the peripheral restriction map; the paper verifies it at $q=-1$ for an infinite class of knots, but the full quantum statement remains open.