Vanishing of the first big Cohen–Macaulay Hilbert coefficient and BCM-rationality

Let $(R,\mathfrak{m})$ be a Noetherian local ring such that its completion $\widehat{R}$ is reduced and satisfies Serre's $S_2$ condition. Let $B$ be a balanced big Cohen–Macaulay $R$-algebra satisfying the condition in the source. For a parameter ideal $Q\subseteq R$, let $e_1^B(Q)$ denote the first Hilbert coefficient associated to $B$, and let $R$ be $\operatorname{BCM}_B$-rational when it is Cohen–Macaulay and the natural map $H^d_{\mathfrak{m}}(R)\to H^d_{\mathfrak{m}}(B)$ is injective, where $d=\dim R$. Vanishing conjecture for the first big Cohen–Macaulay Hilbert coefficient. If $e_1^B(Q)=0$ for some parameter ideal $Q\subseteq R$, then $R$ is $\operatorname{BCM}_B$-rational. In particular, if $R$ is excellent of characteristic $p>0$, its completion is reduced and satisfies $S_2$, and $e_1^*(Q)=0$ for some parameter ideal $Q\subseteq R$, then $R$ is $F$-rational. This conjecture extends Huneke's question beyond the Cohen–Macaulay setting and to all characteristics; the stated condition on $B$ is referenced in the source but not defined in the supplied text, and the resolution status is not established here.