Positivity conjecture for equivariant motivic Chern classes of Schubert cells

Let $G/B$ be a complete flag manifold with Weyl group $W$, and write the equivariant motivic Chern class of a Schubert cell as

$$MC_y^{T}(X(w)^\circ)=\sum_{u\leq w}c_{u,w}(y,e^t)\mathcal O_u^{T}\in K_{\mathbb{T}}(G/B)[y].$$

Positivity conjecture for motivic Chern classes. For every $u\leq w\in W$,

$$(-1)^{\ell(w)-\ell(u)}c_{w,u}(y,e^t)\in {\mathbb Z}_{\geq 0}[y][e^{-\alpha_1},\ldots,e^{-\alpha_r}].$$

Equivalently, the coefficients $(-1)^{\ell(w)-\ell(u)}c_{u,w}(y,e^t)$ should be polynomials in $y$ and $e^{-\alpha_1},\ldots,e^{-\alpha_r}$ with non-negative coefficients. The conjecture is known after specializing $y=0$, and has been checked computationally in the listed low-rank cases, but remains open in general.