Unconditional S-tuple extension of the mean value theorem for quadratic extensions

Let $S$ be a finite set of places of a number field $k$ containing all archimedean places, and let $L_S=(L_v)_{v\in S}$ be an unconditional $S$-tuple of separable quadratic algebras. Unconditional S-tuple conjecture. The statement of Theorem also holds for any unconditional $S$-tuple $L_S$. This would extend the conditional mean value theorem to unconditional local specifications; the source explains that proving it appears to require knowledge of the principal part at the rightmost pole of the global zeta function for ${\EuScript{B}}={\rm M}(2,2)$, which is described as an open problem.