Multiplicity conjecture for Dedekind zeta zeros in non-Galois extensions

Let $L/K$ be an extension of number fields, let $M$ be the Galois closure of $L/K$, and write

$$H=\operatorname{Gal}(L/K)\subset G=\operatorname{Gal}(M/K).$$

If the induced representation $\operatorname{Ind}_H^G(1_H)$ contains an irreducible representation of $G$ with multiplicity greater than one in its decomposition, then $\zeta_L(s)$ has infinitely many nontrivial zeros of multiplicity greater than $1$. Multiplicity conjecture for non-Galois extensions. The stated implication is conjectured as an extension of the results obtained for Galois extensions. It is known in some cases, for example when there exists a field $K\subset K'\subset L$ such that $L/K'$ is a nonabelian Galois extension, but remains open in cases where the relevant subgroup may be maximal.