Tensor-product compatibility conjecture for Koszul complexes and twisted modules

Let $S_n$ be the symmetric group, let $R$ be the polynomial ring used to define the complexes $K_r$, and let $\mathcal{A}_w$ be the dg algebra whose modules are being considered. For a reflection $r=(i,j)$, let

$$K_r=K_{ij}=[R(-1)\xrightarrow{x_i-x_j}\underline{R}(1)]$$

be the associated Koszul complex. Let $X$ be an $\mathcal{A}_w$-module for $w\in S_n$. Tensor-product compatibility conjecture. For any reflection $r$, there should be a weak equivalence of $\mathcal{A}_{wr}$-modules

$$X\otimes_R K_r\cong K_{wrw^{-1}}\otimes_R X.$$

The statement is presented as an expected compatibility and is not needed or proved in the paper. It concerns how tensoring with the Koszul complex associated to a reflection interacts with the twisted module structure.