Nakajima's monomial realization conjecture for highest weight crystals

Let ${\mathcal M}$ be the set of monomials in variables $Y_i(n)$, with $i\in I$ and $n\in\mathbb{Z}$, equipped with the crystal structure defined by the functions and operators in the construction above. Let $M$ be a product of positive powers of the variables $Y_i(n)$, and let $\operatorname{wt}(M)$ denote its weight. Nakajima's monomial realization conjecture. The connected component containing $M$ is isomorphic to the highest weight crystal $B(\operatorname{wt}(M))$. This claim concerns the realization of highest weight crystals through monomials; the surrounding discussion notes that the full set ${\mathcal M}$ does not satisfy all crystal axioms, so the assertion is specifically about components generated by products of positive powers.