The generic Hochschild cohomology conjecture for rank-one double affine Hecke algebras

Let $H(\underline{t};q)$ be the rank-one double affine Hecke algebra with parameters $q$ and $\underline{t}$. Here $HH^i$ denotes its $i$-th Hochschild cohomology group, and $HH^{>2}$ denotes the Hochschild cohomology in all degrees greater than $2$. Generic Hochschild cohomology conjecture. For generic values of $q$ and $\underline{t}$,

$$HH^1(H(\underline{t};q))=HH^{>2}(H(\underline{t};q))=0,\quad HH^2(H(\underline{t};q))=\mathbb C^5.$$

The preceding calculation for the formal deformation at $q=e^h$ and smooth cubic surface suggests this description, but the authors state that they are unable to calculate the Hochschild cohomology directly. The conjecture concerns the generic parameter values and is presented as the possible answer rather than as a proved result.