The degree bound for generators involving the second variable in Jacobian generic initial ideals

Let $\mathcal{A}$ be a central arrangement in $\mathbb{K}^{n+1}$. Write $\operatorname{rgin}(J(\mathcal{A}))$ for the generic initial ideal of its Jacobian ideal, and set

$$p_0=\min\{p\mid x_1^p\in\operatorname{rgin}(J(\mathcal{A}))\}.$$

A minimal generator involving $x_2$ is a minimal monomial generator $t$ of $\operatorname{rgin}(J(\mathcal{A}))$ that contains $x_2$ as a factor. The degree bound conjecture. If $\operatorname{rgin}(J(\mathcal{A}))$ has a minimal generator $t$ involving $x_2$, then

$$\deg(t)\geq p_0.$$

This conjecture concerns the structure of generic initial ideals of Jacobian ideals of central arrangements and would constrain the degrees of generators involving variables beyond $x_1$. The supplied source does not state that it has been proved or disproved, so its status remains open.