On the Frobenius closure of parameter ideals when the ring is F-injective on the punctured spectrum

Let $(R,\mathfrak{m})$ be an excellent generalized Cohen-Macaulay local ring of dimension $d$ that is $F$-injective on the punctured spectrum. Let $\mathfrak{q}$ be a standard parameter ideal of $R$. The aim of the paper is to prove that $${\mathrm{\ell }}_R({\mathfrak{q}}^F\mathrm{/}\mathfrak{q})\le \sum _{i=0}^d\left(\genfrac{}{}{0px}{}{d}{i}\right){\mathrm{\ell }}_R(0_{H_{\mathfrak{m}}^i(R)}^F)\mathrm{.}$$ Moreover, if $\mathfrak{q}$ is contained in a large enough power of $\mathfrak{m}$, we have $${\mathfrak{q}}^F\mathrm{/}\mathfrak{q}\cong \underset{i=0}{\overset{d}{⨁}}(0_{H_{\mathfrak{m}}^i(R)}^F{)}^{\left(\genfrac{}{}{0px}{}{d}{i}\right)}\mathrm{.}$$