Positivity conjecture for the K-theory stable-basis transition coefficients

Let $G/B$ be the flag variety for a semisimple group of rank $n$, let $W$ be its Weyl group, and let $\tau_w^+$ be the coefficients in the expansion of the positive stable basis in the Bott–Samelson basis. Write

$$\tau_w^+=\sum_{v\leq w}d^+_{w,v}Y_v.$$

Here $\ell(w)$ denotes the Weyl-group length, $q$ is the K-theoretic parameter, and $e^{\alpha_1},\ldots,e^{\alpha_n}$ are the exponentials of the simple roots. Positivity conjecture. The coefficients $d^+_{w,v}$ satisfy

$$d_{w,v}^+=(-1)^{\ell(w)}\sum_{i=0}^{\ell(v)}(-q)^i u_i,$$

where $u_i$ are polynomials in $\mathbb{Z}[e^{\alpha_1},\ldots,e^{\alpha_n}]$ with positive coefficients. This conjecture predicts a sign-alternating, positive-coefficient expansion for the transition matrix between Schubert classes and positive K-theory stable bases; the source provides no resolution, so its status remains open.