Equivariant Seidel multiplication conjecture in quantum K-theory

Let $X=G/P$ be a flag variety over $\mathbb C$, let $W^{\mathrm{comin}}$ be the subgroup of point representatives of cominuscule flag varieties together with the identity, and let $\operatorname{QK}_T(X)$ be the equivariant quantum $K$-theory ring. For $w\in W^{\mathrm{comin}}$ and $u\in W$, let $d(w,u)$ be the associated effective curve degree. Equivariant Seidel multiplication conjecture.

$$[\mathcal O_{X_{w_0w}}]\star[\mathcal O_{X^u}]=q^{d(w,u)}[\mathcal O_{w^{-1}.X^{wu}}]$$

and

$$[\mathcal O_{X^w}]\star[\mathcal O_{w.X^u}]=q^{d(w,u)}[\mathcal O_{X^{wu}}]$$

in $\operatorname{QK}_T(X)$. The two identities are equivalent. The non-equivariant case is known for cominuscule flag varieties, while the equivariant statement is the quantum $K$-theoretic analogue of the established quantum-cohomology Seidel formula.