The q-deformed random-to-random commutativity conjecture

Assume $q\neq0$. Let $\operatorname{\mathcal{H}}_n^{\operatorname{aff}}$ be the affine Hecke algebra containing $\operatorname{\mathcal{H}}_n$ and the Laurent polynomial ring in $x_1,\ldots,x_n$, and define

$$\operatorname{\widetilde{\mathcal{R}}}_{n,k}:=\frac{1}{[k]!_q}\operatorname{\mathcal{B}}^*_{n,k}(x_{n-k+1}\cdots x_{n-1}x_n)\operatorname{\mathcal{B}}_{n,k}\in\operatorname{\mathcal{H}}_n^{\operatorname{aff}}.$$

Here $n,k\geq0$, $[k]!_q$ is the $q$-factorial, and $\operatorname{\mathcal{B}}_{n,k}$ is the element used in this definition. The q-deformed commutativity conjecture. For every $n\geq0$, the elements $\operatorname{\widetilde{\mathcal{R}}}_{n,k}$ commute for all $k\geq0$. This is proposed as an analogue of the corresponding commutativity theorem for the undeformed elements; the affine-Hecke-algebra deformation is presented as conjectural, and no result resolving it is given.