Quantum nonnegative monomial-basis conjecture for crystallographic Coxeter systems

Let $(W,S)$ be a crystallographic Coxeter system, let $qBE(W,S)$ be the associated quantum algebra, and let $\tilde{P}_w\in {\bf S}(V^{\ast})\otimes {\bf R}[q_s]$ be the polynomial characterized by $[\tilde{P}_w](1)=w$ and by its triangular expansion in the polynomials $X_v$. Quantum nonnegative monomial-basis conjecture. There exists a monomial basis $\{b_{\mu}\}_{\mu}$ of $qBE(W,S)$ such that, for every $w\in W$, the element $[\tilde{P}_w]$ is a linear combination of the basis elements $b_{\mu}$ with nonnegative coefficients that do not depend on the parameters $q_s$. This is the quantum analogue of the preceding positivity conjecture; the supplied passage gives no resolution status or further evidence.