The root-difference conjecture for free hyperplane arrangements

Let ${\mathcal{A}}$ be a free hyperplane arrangement with exponents

$$\exp({\mathcal{A}})=(d_1,d_2,\ldots,d_\ell)_\le,$$

where $d_1=1$, and let $H\in{\mathcal{A}}$. Write ${\mathcal{A}}^H$ for the restriction to $H$ and set

$$d:=|{\mathcal{A}}|-|{\mathcal{A}}^H|.$$

Root-difference conjecture. Then $d=d_i$ for some $i$, or $d>d_\ell$. The source notes that this conclusion was established under the additional assumption that ${\mathcal{A}}\setminus\{H\}$ is locally free, and conjectures that the same conclusion holds without that assumption.