The modular-flat vanishing conjecture for hyperplane-arrangement forms

Let $\mathcal{K}$ be a hyperplane arrangement, let $F$ be a flat, let $J$ be a set of hyperplanes, let $\iota$ be as above, and let $\omega_{\mathcal{K},\iota,F,J}$ denote the associated form. A flat $F'$ is proper modular when it is a proper modular flat of $\mathcal{K}$; assume that $F'\subseteq F$. Modular-flat vanishing conjecture. If $J$ contains exactly $2$ hyperplanes not containing $F'$, then

$$\omega_{\mathcal{K},\iota,F,J}=0.$$

The conjecture extends the preceding vanishing argument, which handles the case where $J$ contains a unique hyperplane not containing $F'$. The source does not give a resolution of the general case.