Log-concavity conjecture for Milnor algebra local cohomology

Let $S$ be the graded polynomial ring defining a projective hypersurface $V(f):f=0$, and let $M(f)=S/J_f$ be its Milnor algebra, where $J_f$ is the Jacobian ideal. Let $N(f)=H^0_{\mathbf m}(M(f))=\widehat J_f/J_f$, with $\mathbf m$ the maximal ideal and $\widehat J_f$ the saturation of $J_f$. Assume that $V(f)$ has only isolated singularities, so that $HP(N(f))(t)$ is a polynomial. Log-concavity conjecture. The series $HP(N(f))(t)$ is a log-concave polynomial with no internal zeros. In particular, $HP(N(f))(t)$ is a unimodal polynomial. This conjecture strengthens Dimca's conjecture and concerns the replacement of smooth-hypersurface properties of the Milnor algebra by properties of the local cohomology module $N(f)$; the isolated-singularity hypothesis ensures that its Hilbert–Poincaré series is finite.