Nonnegative monomial-basis conjecture for Bernstein–Eilenberg algebras

Let $(W,S)$ be a crystallographic Coxeter system. Let $BE(W,S)$ be the associated algebra, let $X_w$ be the Bernstein–Gelfand–Gelfand polynomial indexed by $w\in W$, and let $[X_w]$ denote its image in $BE(W,S)$. Nonnegative monomial-basis conjecture. There exists a monomial basis $\{b_{\mu}\}_{\mu}$ of $BE(W,S)$ such that, for every $w\in W$, the element $[X_w]$ is a linear combination of the basis elements $b_{\mu}$ with nonnegative coefficients. The claim proposes a positivity refinement for the images of Schubert-type polynomials in $BE(W,S);$ the supplied passage gives no resolution status or further evidence.