Characterization of Jacobian linear type for hypersurfaces with one-dimensional singularities

Let $X=V(f)\subseteq \mathbb{A}_k^n$ be an affine hypersurface with one-dimensional singularities. For a minimal prime $\mathfrak q$ of the Jacobian ideal $I(f)$, let $T(f)_{\mathfrak q}$ and $M(f)_{\mathfrak q}$ denote the local Tjurina and Milnor algebras, respectively, and let $\ell(-)$ denote length. Jacobian linear type characterization. The hypersurface $X$ is of Jacobian linear type if and only if, for each minimal prime $\mathfrak q$ of $I(f)$,

$$\ell\bigl(T(f)_{\mathfrak q}\bigr)=\ell\bigl(M(f)_{\mathfrak q}\bigr)$$

and $I(f)_{\mathfrak q}$ is a complete intersection. This proposes a length-theoretic and local complete-intersection criterion for Jacobian linear type in the one-dimensional singular-locus case; the source presents it as a question motivated by the failure of analogous behavior for higher-dimensional singularities, and gives no resolution.