The standard and costandard module conjecture for non-symmetric quantum loop groups

For each $w\in\mathbb{N} I^\bullet$, let $W$ be the corresponding framing data, and let $\widetilde{\mathfrak{M}}^\bullet(W)$ be the associated moduli space with potential $\widetilde{f}^\bullet_1$. Write $K(\widetilde{\mathfrak{M}}^\bullet(W),\widetilde{f}^\bullet_1)$ for its critical K-theory module and $K(\widetilde{\mathfrak{M}}^\bullet(W),\widetilde{f}^\bullet_1)_{\widetilde{\mathfrak{L}}^\bullet(W)}$ for the corresponding module supported on $\widetilde{\mathfrak{L}}^\bullet(W)$. Standard and costandard module conjecture. For each $w\in\mathbb{N} I^\bullet$, the $\mathrm{U}_\zeta(\mathrm{L}\mathfrak{g})$-modules

$$K(\widetilde{\mathfrak{M}}^\bullet(W),\widetilde{f}^\bullet_1),\quad K(\widetilde{\mathfrak{M}}^\bullet(W),\widetilde{f}^\bullet_1)_{\widetilde{\mathfrak{L}}^\bullet(W)}$$

are isomorphic, respectively, to the costandard module and the standard module with $\ell$-highest weight $\Psi_w$. This conjecture identifies the critical K-theory modules constructed from the moduli spaces with the standard and costandard representations of the non-symmetric quantum loop group; the supplied text gives no resolution, so its status is open.