Integrable extension conjecture for the $\widehat{\mathfrak{sl}}_n$-module $V(\mu)$

Let $V(\mu)$ be the $\mathbb C$-linear span of the vectors associated with the allowed collections of fixed-point data of weight $\mu$. Regard $V(\mu)$ as a module over $\widehat{\mathfrak{sl}}_n$ by restricting the Yangian action.

Integrable extension conjecture. The $\widehat{\mathfrak{sl}}_n$-module $V(\mu)$ extends to the irreducible integrable $\widehat{\mathfrak{gl}}_n$-module with highest weight $\mu$.

This predicts that the restricted module admits the expected integrable affine general linear extension. The supplied text gives no resolution status.