Maximal analytic spread implies linear type for linear free divisors

Let $R=k[x_1,\cdots,x_n]$ be a polynomial ring, let $f\in R$ be a linear free divisor, and let $J_f$ denote its Jacobian ideal. Write $\ell(J_f)$ for the analytic spread of $J_f$.

Maximal-spread linear-type conjecture. If

$$\ell(J_f)=n,$$

then $J_f$ is of linear type.

The conjecture concerns the relationship between maximal analytic spread and the linear-type property; the paper identifies $n\geq 5$ as the main case of interest, while the supplied text gives no resolution.