The WKB crystal structure conjecture for shift-of-argument eigenlines

Let $L(\lambda)$ be a finite-dimensional representation of $U(\mathfrak{gl}_n)$, let $u\in\widehat{\mathfrak h_{\rm reg}}(\mathbb{R})$, and let $E(\lambda,u)$ be the set of eigenlines for the shift-of-argument subalgebra $\mathcal{A}(u)$. For each $k=1,\ldots,n-1$, denote by $s_{k,k+1}^{(+)}(u)$ and $s_{k+1,k}^{(-)}(u)$ the relevant operators, and by $\tilde e_k(u)$ and $\tilde f_k(u)$ their WKB-induced operators. WKB crystal structure conjecture. The WKB approximations of $s_{k,k+1}^{(+)}(u)$ and $s_{k+1,k}^{(-)}(u)$ produce operators $\tilde e_k(u)$ and $\tilde f_k(u)$ on $E(\lambda,u)$ such that $(E(\lambda,u),\tilde e_k(u),\tilde f_k(u))$ is a $\mathfrak{gl}_n$-crystal. Moreover, for any two representations $L(\lambda_1)$ and $L(\lambda_2)$, the WKB approximations of the displayed tensor-product operators in the source produce the crystal operators $\tilde e_k(u)$ and $\tilde f_k(u)$ on the tensor product $E(\lambda_1,u)\otimes E(\lambda_2,u)$ of $\mathfrak{gl}_n$-crystals. This conjecture proposes that the WKB asymptotics of shift-of-argument operators recover the crystal structure on eigenlines and are compatible with tensor products.