Quadratic growth of maximal Schubert regularity

For each $n$, define

$$\operatorname{maxReg}(n)=\max_{v\leq w\in\mathfrak S_n}\operatorname{Reg}(R_{v,w}),$$

where $R_{v,w}$ is the associated graded ring of the local ring of the Schubert variety $X_w$ at the torus-fixed point $e_v$. Maximal-regularity conjecture. The maximal regularity has quadratic order of growth:

$$\operatorname{maxReg}(n)=\Theta(n^2).$$

This predicts quadratic asymptotic growth for the largest regularity among Schubert tangent cones in type $A$. The source gives no resolution status or supporting evidence beyond the conjectural assertion.