Tight closure of parameter ideals in local rings $F$-rational on the punctured spectrum

Let $(R,\mathfrak{m},k)$ be an excellent equidimensional local ring of characteristic $p>0$. The aim of this paper is to show that ${\mathrm{\ell }}_R({\mathfrak{q}}^{\ast }\mathrm{/}\mathfrak{q})$ does not depend on the choice of parameter ideal $\mathfrak{q}$ provided $R$ is an $F$-injective local ring that is $F$-rational on the punctured spectrum.