Hassanzadeh–Nejad–Simis conjecture on the Jacobian colon ideal

Let $k$ be a field of characteristic zero and let

$$R=k[[x_1,\dots,x_n]].$$

Let $f\in R$ define an isolated hypersurface singularity, with Jacobian ideal $j(f)$ and Tjurina ideal $tj(f)$. Hassanzadeh–Nejad–Simis conjecture.

$$j(f):f\nsubseteq tj(f).$$

This conjecture was raised as Conjecture 3.6 in the cited work and is presented here without a resolution.