Determinacy of smooth germs with real isolated line singularities

The germ of a smooth real-valued function on Euclidean space is called a real isolated line singularity if its singular set is a nonsingular curve, its Jacobian ideal is ojasiewicz at the singular set, and its Hessian determinant restricted to the singular set is ojasiewicz at 0. Consider the set of all germs whose singular set contains a fixed nonsingular curve $L$. We prove that such a germ $f$ is infinitely determined among all such germs with respect to composition by diffeomorphisms preserving $L$ if, and only if, the Jacobian ideal of $f$ contains all germs which vanish on $L$ and are infinitely flat at 0 if, and only if, $f$ is a real isolated line singularity.