Exact formula conjecture for the minimal total degree of flag-variety regularity

Let ${\mathcal{F}}l(\mathbb C^n)$ be the complete flag variety, and let $\operatorname{reg}(\iota_*{\mathcal{O}}_{{\mathcal{F}}l(\mathbb C^n)})$ be its multigraded regularity region. Define

$$m(n)=\min\{\lvert \mathbf a\rvert:\mathbf a\in\operatorname{reg}(\iota_*{\mathcal{O}}_{{\mathcal{F}}l(\mathbb C^n)})\},$$

where $\lvert\mathbf a\rvert=\sum_i a_i$. Exact-degree conjecture. For every $n\geq4$,

$$m(n)=\binom{n-2}{2}.$$

This is a precise form of the proposed quadratic growth, motivated by the computed regularity regions in low dimensions; proving it would sharpen the currently known linear lower and cubic upper bounds.