Quantum K-theoretic Seidel operator and Pieri rule for complete flag varieties

Let $F\ell_n$ be the complete flag variety, let $S_n$ be its Weyl group with simple reflections $s_1,\ldots,s_{n-1}$, and let $QK(F\ell_n)$ denote its quantum $K$-theory with quantum parameters $q_1,\ldots,q_{n-1}$. For $u\in S_n$, write $k:=n-u(n)$, and use the operations $u\uparrow r$, $\lambda(u)$, $\lambda(u,k)$, and $\mathcal{T}$ as in the displayed formulas. Quantum Seidel–Pieri conjecture. For every $1\leq m\leq n-1$ and $u\in S_n$, one has

$$\mathcal{T}(\mathcal{O}^u):=\mathcal{O}^{s_1\cdots s_{n-2}s_{n-1}}*\mathcal{O}^u=q_{\lambda(u)}\mathcal{O}^{u\uparrow 1},$$

and

$$\mathcal{O}^{s_{n-m}\cdots s_{n-2}s_{n-1}}*\mathcal{O}^u=q_1^{-1}q_2^{-2}\cdots q_{n-1}^{1-n}q_{\lambda(u,k)}\mathcal{T}^{n-k}\bigl(\mathcal{O}^{s_{n-m}\cdots s_{n-2}s_{n-1}}\cdot\mathcal{O}^{u\uparrow k}\bigr).$$

The conjecture proposes quantum $K$-theoretic analogues of the stated Seidel-element formula and quantum Pieri rule for $QH^*(F\ell_n)$; the preceding discussion notes that the quantum $K$-theoretic structure constants agree with their cohomological counterparts in the indicated degree, but no resolution of these formulas is supplied here.