The secondary Euler characteristic equality conjecture for totally real fields

Let $k$ be a totally real number field, let $p$ be a prime, let $E$ be a finite extension of $\mathbb Q_p$, and let $\chi$ be an even Artin character of $G_k$ realizable over $E$. Let $S$ contain the primes above $p$ and the finite primes where $k_\chi/k$ is ramified. For $e\in\mathfrak{B}_k(E)$, write $\nu_S(1-e,\chi)$ for the order of vanishing of the relevant $p$-adic $L$-function and let $\operatorname{cork}_{i,S}(1-e,\chi)$ denote the corresponding corank. The secondary Euler characteristic equality conjecture. We have

$$\nu_S(1-e,\chi)=\operatorname{cork}_{2,S}(1-e,\chi)-\operatorname{cork}_{0,S}(1-e,\chi).$$

The equality is independent of the choice of $S$; the paper explains that it would follow from certain folklore conjectures in Iwasawa theory and gives reductions via induction and Brauer induction, but it remains unproved in general.