The geometric Razumov–Stroganov conjecture for conormal Schubert sheaves

Let $N=2n$, let $LP(N)$ be the set of full link patterns, and let $\Psi_r$ be the polynomial indexed by $r\in LP(N)$ from the wheel-condition basis. Let $\operatorname{Mat}_<(N)$ be the affine space of strict upper-triangular $N\times N$ matrices, and let $\mu$ be the proper map from the union of conormal Schubert varieties to $\operatorname{Mat}_<(N)$. Write $\tikz[scale=0.3]{\draw (0,0) -- (1,1) -- (0,1) -- cycle;}$ for the staircase link-pattern region and let $\tilde m_r$ be the normalization monomial.

Geometric Razumov–Stroganov conjecture. The pushforward of $\sigma_r$ to $\operatorname{Mat}_<(N)$ in $T$-equivariant $K$-theory is equal, up to normalization, to

$$\mu_*[\sigma_r]=\begin{cases}0&r\not\subseteq\tikz[scale=0.3]{\draw (0,0) -- (1,1) -- (0,1) -- cycle;},\\(1-t)^{n(n-1)}\tilde m_r\,\Psi_r&r\subseteq\tikz[scale=0.3]{\draw (0,0) -- (1,1) -- (0,1) -- cycle;}.\end{cases}$$

This conjecture identifies geometric pushforwards of conormal Schubert sheaves with the polynomial solutions of the level-one quantum Knizhnik–Zamolodchikov system. The supplied text does not state that it has been proved or disproved.