Quantum K-theory positivity conjecture for homogeneous spaces

Let $X=G/P_X$ be a rational projective homogeneous space. Schubert varieties in $X$ are indexed by $W^X$, and for $u\in W^X$ let $X^u$ be the Schubert variety with $\operatorname{codim}_X(X^u)=\ell(u)$. Write ${\mathcal O}^u=[{\mathcal O}_{X^u}]$ for its class in the small quantum $K$-theory ring $\operatorname{QK}(X)$, whose products are

$${\mathcal O}^u\star {\mathcal O}^v=\sum_d\sum_{w\in W^X}N_{u,v}^{w,d}q^d{\mathcal O}^w,$$

where $d$ ranges over effective classes in $H_2(X,\mathbb Z)$. Quantum $K$-theory positivity conjecture. For $u,v,w\in W^X$ and $d\in H_2(X,\mathbb Z)$, one has

$$(-1)^{\ell(uvw)+\int_d c_1(T_X)}N_{u,v}^{w,d}\geq 0.$$

This conjecture extends the known sign alternation for structure constants in ordinary and classical $K$-theory to quantum $K$-theory of homogeneous spaces. The source presents it as an expected positivity property and gives no resolution of the general statement.