Equality of the jet Tjurina index and nilpotency index

Let $k$ be an algebraically closed field of characteristic $0$, let $f\in k[[x_{1},...,x_{n}]]$ define an isolated singularity at $0$, and let $J(f)$ be the Jacobian ideal of $f$. Let $j_{\tau}(f)$ denote the jet Tjurina index, and let $N(f,J(f))$ be the corresponding nilpotency index, so that the notation refers to the index introduced immediately before the conjecture. Jet Tjurina index conjecture.

$$j_{\tau}(f)+1=N(f,J(f)).$$

The paper notes the general inequality $j_{\tau}(f)+1\geq N(f,J(f))$ and reports that extensive calculations found no counterexample to equality. The conjecture is presented without a proof.