Pollio–Rapinchuk's multinorm principle conjecture

Let $k$ be a field and let $L_1$ and $L_2$ be finite Galois extensions of $k$. An extension $P$ of $k$ satisfies the norm principle when the associated norm principle holds, and the pair $L_1,L_2$ satisfies the multinorm principle when the associated multinorm principle holds.

Pollio–Rapinchuk's conjecture. If every extension $P$ of $k$ contained in $L_1\cap L_2$ satisfies the norm principle, then the pair $L_1,L_2$ satisfies the multinorm principle. It may be enough to require only that the intersection $L_1\cap L_2$ satisfies the norm principle.

The conjecture concerns the Hasse principle for multinorm equations and relates it to norm principles for the subextensions contained in the intersection of the two Galois extensions. The paper proves an analogue for weak approximation under technical assumptions, but the supplied text does not establish this conjecture itself.