Huneke's conjecture on tight Hilbert coefficients and F-rationality

Let $(R,\frak{m})$ be an excellent Noetherian local ring of characteristic $p>0$ such that its completion $\widehat{R}$ is reduced and satisfies $(S_2)$. For a parameter ideal $Q\subseteq R$, let $e_1^*(Q)$ denote the first tight Hilbert coefficient. Huneke's conjecture. If

$$e_1^*(Q)=0$$

for some, and hence for all, parameter ideals $Q\subseteq R$, then $R$ is $F$-rational. The conjecture asks whether vanishing of the first tight Hilbert coefficient characterizes $F$-rationality under these hypotheses; the paper's abstract indicates that this implication is proved for Buchsbaum local rings satisfying $(S_2)$, while the general case remains the question posed by Huneke.