Flat-locus dimension conjecture for linear degenerations of Schubert varieties

Let $X_w$ be a Schubert variety, let $R^{\iota}_{\bf d}$ be the relevant representation variety, let $Y$ be the universal quiver Grassmannian, and let

$$\pi:Y\to R^{\iota}_{\bf d}$$

be its projection. For $f\in R^{\iota}_{\bf d}$, write $M^f$ for the corresponding representation and $\operatorname{Gr}_{{\bf e}^w}(M^f)$ for its quiver Grassmannian.

Flat-locus dimension conjecture. A tuple $f\in R^{\iota}_{\bf d}$ is in the flat locus of $\pi$ if and only if

$$\dim\bigl(\operatorname{Gr}_{{\bf e}^w}(M^f)\bigr)=\dim(X_w).$$

This conjecture would give a dimension-theoretic characterization of the flat fibres in the family of linear degenerations of Schubert varieties. The source presents it as a potential strategy and motivation; no resolution is supplied.