The flagged Schur module character conjecture for northwest diagrams

Let $D$ be a northwest diagram. The flagged Schur module $\mathcal{S}^{\mathrm{flag}}_{D}$ has character $\mathrm{char}(\mathcal{S}^{\mathrm{flag}}_{D})$, and the Kohnert polynomial of $D$ is denoted by $\mathfrak{K}_{D}$.

Flagged Schur module character conjecture. For every northwest diagram $D$,

$$\mathrm{char}(\mathcal{S}^{\mathrm{flag}}_{D})=\mathfrak{K}_{D}.$$

This conjecture predicts that the character of each flagged Schur module is given by the Kohnert polynomial of its northwest diagram, connecting flagged Schur modules with Kohnert diagrams and their Demazure-character expansions. The supplied text does not state whether the conjecture has been resolved.