Product monomial crystal conjecture for highest weights of truncated shifted Yangians

Let $\mathfrak g$ be a semisimple simply-laced Lie algebra, let $\lambda$ and $\mu$ be dominant coweights, let $Y^\lambda_\mu(\mathbf R)$ be the corresponding truncated shifted Yangian, and let $H^\lambda_\mu(\mathbf R)$ be the set of highest weights $J$ for which the Verma module $M^\lambda_\mu(J,\mathbf R)$ is nonzero. Let $\mathcal B(\lambda,\mathbf R)$ be the product monomial crystal and $\mathcal B(\lambda,\mathbf R)_\mu$ its $\mu$-weight set; for $J\in H^\lambda_\mu(\mathbf R)$, let $y(J)$ be the associated Nakajima monomial. Product monomial crystal conjecture. For any dominant coweights $\lambda,\mu$, the map

$$J\longmapsto y(J)$$

is a bijection

$$H^\lambda_\mu(\mathbf R)\xrightarrow{\sim}\mathcal B(\lambda,\mathbf R)_\mu.$$

The conjecture describes the possible highest weights of truncated shifted Yangians in terms of natural subcrystals of Nakajima's monomials. It is proved when $\mathfrak g=\mathfrak{sl}_n$ and, when $\lambda$ is a multiple of the first fundamental weight, follows from work of Brundan and Kleshchev; the general simply-laced case remains open.