The key conjecture on quantum group representation rings and DAHA

Let $G$ be a quantum group with Lusztig's quantum group, let $Rep_q\,G$ denote its category of finite-dimensional representations, and let $K_0(Rep_q\,G)$ be its Grothendieck algebra. Let $W$ be the Weyl group of the corresponding root system, and let the polynomial representation of the double affine Hecke algebra (DAHA) be specialized at $t=q$.

Key conjecture. The commutative algebra $K_0(Rep_q\,G)$ can be canonically identified with the algebra of $W$-invariants in this polynomial representation of DAHA. For generic $q$, the simple objects correspond to the characters, namely the eigenfunctions of the $Y$-operators, and the fusion procedure becomes multiplication.

This conjecture proposes a connection between the representation theory of Lusztig's quantum groups and DAHA within the expected relationship between Langlands-program structures and DAHA. The supplied context does not state whether the conjecture has been proved or disproved.