The Euler conductor noncontainment conjecture for isolated hypersurface singularities

Let $R=k[[x_1,\ldots,x_n]]$ define an isolated hypersurface singularity, with $f\in R$ and $k$ a field of characteristic zero. Let $J_f$ be the Jacobian ideal and let $I_f$ denote the ideal used in the paper's definition of the Euler conductor. Euler conductor noncontainment conjecture. One has

$$J_f:f\not\subset I_f.$$

The paper reports that no evidence has been found for a nonzero invariant $\beta_f$, and formulates this noncontainment as a conjectural assertion for isolated hypersurface singularities.