The BMR freeness and symmetrizing-form conjecture for cyclotomic Hecke algebras

Let $W$ be an irreducible complex reflection group, let $A$ be the coefficient ring, and write ${\mathcal{H}}(W,{\mathbf u})$ for its cyclotomic Hecke algebra. Let $B(W)$ be the braid group, let $\boldsymbol\pi$ be the element defined from a generator of the center as in the source, and write $b\mapsto T_b$ for the natural map from $B(W)$ to ${\mathcal{H}}(W,{\mathbf u})$. Let $x\mapsto x^\vee$ be the automorphism of $A$ given by ${\mathbf u}\mapsto{\mathbf u}^{-1}$. The BMR freeness and symmetrizing-form conjecture.

(a) ${\mathcal{H}}(W,{\mathbf u})$ is free over $A$ of rank $|W|$.

(b) ${\mathcal{H}}(W,{\mathbf u})$ carries a non-degenerate symmetrizing form

$$t:{\mathcal{H}}(W,{\mathbf u})\longrightarrow A$$

which makes it into a symmetric algebra and satisfies

$$t(T_{b^{-1}})^\vee=\frac{t(T_{b\boldsymbol\pi})}{t(T_{\boldsymbol\pi})}\qquad\text{for all }b\in B(W).$$

This is presented as a stronger assertion following the basic dimension and representation conjecture. The source notes that the conjecture is known only in some cases, so it remains open in general.