Vanishing of the $4\mu4$-invariant after totally real base change

Let $F$ be a totally real number field, let $K$ be a quadratic imaginary extension of $F$, let $F'$ be a finite totally real extension of $F$, and let $f_{F'}$ be a Hilbert modular form over $F'$ that is ordinary at each place of $F'$ above $p$. $4\mu4$-vanishing conjecture. One has

$$\mu\bigl(L_p(F'K_\infty,f_{F'})\bigr)=0.$$

This predicts the absence of a nontrivial $4p4$-power factor in the anticyclotomic $p$-adic $L$-function after arbitrary finite totally real base change; the paper places it in relation to a preceding $4\mu4$-vanishing corollary, but the general assertion remains conjectural in the supplied text.