Existence of indecomposable graded bimodules realizing the Kazhdan–Lusztig basis

Let $V$ be a reflection-faithful representation of ${\cal W}$ over an infinite field, and let $R$ be the ring of regular functions on $V$. For $x\in{\cal W}$, let $C'_x$ be the corresponding element of the Kazhdan–Lusztig basis and let ${\cal E}$ be the ring homomorphism from the Hecke algebra to the Grothendieck ring of graded $R$-bimodules described above. Bimodule-realization conjecture. At least when $k=\Bbb{C}$, there exists an indecomposable $\mathbb{Z}$-graded $R$-bimodule $B_x\in{\cal R}$ such that

$${\cal E}(C'_x)=\langle B_x\rangle.$$

This asserts that the Kazhdan–Lusztig basis elements are realized by indecomposable graded bimodules. The supplied text gives no resolution status beyond the qualification that the assertion is expected at least over $\mathbb{C}$, so it is recorded as open.