The essential-set containment conjecture for simply-laced root systems

Let $\Delta$ be a simply-laced root system with highest root $\theta$, and let $\theta$ be fundamental. For a simple root $\alpha$, let $\EuScript F_{\alpha}$ be the associated minimal inversion complete set, let $I(\alpha)_{\max}$ be its maximal abelian ideal, and let $\mathsf{Ess}(\EuScript F_{\alpha})$ denote its essential set. Let $\mathcal H$ be the set denoted by ${\EuScript H}$ in the source. Essential-set containment conjecture.

$$\bigl(\mathsf{Ess}(\EuScript F_{\alpha})\setminus I(\alpha)_{\max}\bigr)\subset \mathcal H.$$

This conjecture concerns the remaining cases for the series $\mathrm{D}_n$ and $\mathrm{E}_n$ in which the highest root is fundamental; the paper reports only partial computations supporting it, so its general validity remains open.