The flagged skew tableau realization conjecture for Demazure crystals

A Demazure crystal is denoted by $\mathcal{B}_{w,\lambda}$, where the crystal is of type $A$, normal, and finite. A flagged skew tableau crystal is the crystal of flagged skew tableaux of shape $\lambda/\mu$ with a nondecreasing flag $\vec{b}$. Flagged skew tableau realization conjecture. Every Demazure crystal $\mathcal{B}_{w,\lambda}$ is isomorphic to a flagged skew tableau crystal for some shape $\lambda/\mu$ and nondecreasing flag $\vec{b}$. Consequently, all results in the appendix would hold for every Demazure crystal. The statement would extend the appendix's results from tableau-like crystals satisfying row-flagging conditions to all finite normal type $A$ Demazure crystals; the source presents it as an open problem and notes that no reference for the asserted realization was found.