Itoh's conjecture on the third normal Hilbert coefficient

Let $(R,\mathfrak m)$ be an analytically unramified Gorenstein local ring of dimension $d\geq 3$, and let $I$ be a parameter ideal. The normal Hilbert coefficients $\overline{e}_3(I)$ and the normal reduction number $\overline{r}(I)$ are defined from the normal filtration $\{\overline{I^n}\}_{n\geq 0}$, where $\overline{I^n}$ denotes the integral closure of $I^n$. Itoh's conjecture.

$$\overline{e}_3(I)=0 \quad\Longleftrightarrow\quad \overline{r}(I)\leq 2.$$

This conjecture proposes a precise equivalence between vanishing of the third normal Hilbert coefficient and a bound on the normal reduction number. The source describes it as open since 1992; the paper proves related sufficient criteria and generalizations for other normal Hilbert coefficients, but does not resolve this statement.