The Jacobian-colon-integral-closure conjecture for isolated hypersurface singularities

Let $K$ be a field of arbitrary characteristic and let

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

Let $f\in R$ define an isolated hypersurface singularity, let $j(f)$ be its Jacobian ideal, and write $j(f):f$ for the ideal quotient and $\overline{j(f)}$ for the integral closure of $j(f)$. Assume that $f\in\overline{j(f)}$. Jacobian-colon-integral-closure conjecture.

$$j(f):f\nsubseteq\overline{j(f)}.$$

The authors motivate this by noting that they have found no examples indicating the opposite inclusion; the conjecture would imply a sharper bound for the quotient of Milnor number by Tjurina number when $n\geq3$.