The mixed-characteristic colon-capturing conjecture for tight closure

Let $(R,\operatorname{\mathfrak{m}})$ be a $d$-dimensional complete local domain of mixed characteristic $p$. Assume that $p,x_2,\dots,x_d$ is a system of parameters. Set

$$\overline{R}:=R/pR,\qquad J=(x_2,\ldots,x_d)R,$$

and, for $n>0$, define

$$J_n=(x_2,\dots,x_d):_R p^n\subseteq R.$$

Here $(J\overline R)^*$ denotes the tight closure of $J\overline R$ in $\overline R$. Mixed-characteristic colon-capturing conjecture. For all $n>0$,

$$J_n\overline R\subseteq (J\overline R)^*.$$

The text says that affirmative answers to preceding questions about robust closure would imply this conjecture. Its resolution status is not given.