Complete-intersection conjecture for equidimensional radical Jacobian ideals of depth at least three

Let $D$ be a divisor in a complex manifold $S$, defined locally at a point $p$ by a reduced $h \in \mathcal{O}_{S,p}$. Suppose that the Jacobian ideal $J_h$ is radical, equidimensional, and has depth at least $3$ in $\mathcal{O}_{S,p}$. Higher-depth Jacobian conjecture. The variety $\operatorname{Sing} D$ with coordinate ring $\mathcal{O}_{S,p}/J_h$ is a complete intersection; equivalently, $\operatorname{Sing} D$ is Cohen--Macaulay and, by the cited proposition, is smooth. The authors state that non-complete-intersection radical Jacobian ideals of height at least $3$ are not understood and that even their existence is unclear, so this assertion remains open.