Burity–Simis–Tohăneanu minimal-reduction conjecture for Jacobian ideals

Let $k$ be a field, let $R=k[x_1,\ldots,x_n]$, and let $\mathcal A$ be a central hyperplane arrangement with defining linear forms whose product is $f$. Let $I\subseteq R$ be the ideal generated by the $(m-1)$-fold products of the defining linear forms, and let $J_f$ be the Jacobian ideal of $f$. Burity–Simis–Tohăneanu's minimal-reduction conjecture. With this notation, $J_f$ is a minimal reduction of $I$, with reduction number

$$\operatorname{red}_{J_f}(I)\leq n-1.$$

This conjecture was posed by Burity, Simis and Tohăneanu and has been proved when the arrangement has rank at most $3$, including almost generic arrangements.