Radicality conjecture for patches of semisimple Hessenberg varieties

Let $\mathcal{B}(S,H)$ be a semisimple Hessenberg variety in $GL_n(\mathbb C)/B$, and let $I_{w,\mathcal{B}(S,H)}$ be the ideal defined above for $w\in W$. The ideal is radical if it equals its radical.

Radicality conjecture. The ideal $I_{w,\mathcal{B}(S,H)}$ is radical for any semisimple Hessenberg variety $\mathcal{B}(S,H)$, and is therefore the patch ideal of $\mathcal{B}(S,H)$ at $wB$.

The claim would establish that the explicitly defined ideals give the scheme-theoretic patch ideals for all semisimple Hessenberg varieties. It has been verified in the paper for $n\leq 5$, but no general proof is known.