Equality of the relative and dual F-rational signatures

Let $(R,\mathfrak m)$ be an F-rational local ring. Write $\operatorname{s_{rel}}(R)$ for its relative F-rational signature and $\operatorname{s_{dual}}(R)$ for its dual F-rational signature. For an ideal generated by a system of parameters $\underline{x}$, let $\operatorname{e}_{HK}(\langle\underline{x}\rangle)$ denote its Hilbert–Kunz multiplicity, and let $\ell(-)$ denote length. Equality conjecture. One has

$$\operatorname{s_{rel}}(R)=\operatorname{s_{dual}}(R).$$

Moreover, there exists a system of parameters $\underline{x}$ and an ideal $J$ with $\underline{x}\subset J$ such that

$$\operatorname{s_{dual}}(R)=\frac{\operatorname{e}_{HK}(\langle\underline{x}\rangle)-\operatorname{e}_{HK}(J)}{\ell(R/\langle\underline{x}\rangle)-\ell(R/J)}.$$

The paper has already established that the relative, Cartier, dual, and ordinary F-signatures are related by a chain of inequalities, while examples show that the relative signature can differ from the rational signature. The conjecture asserts that the remaining invariants coincide in the F-rational local setting and gives a stronger formula for the dual signature.