Conjectured negative answer to the degree problem via equal admissible groups

Let the degree problem ask whether equivalent number fields must have the same degree. Degree-problem conjecture. The degree problem has a negative answer. Moreover, there exist a number field $M$ and a proper subfield $K\subsetneq M$ such that $K$ and $M$ have the same admissible groups.

This would produce a nontrivial pair of number fields related by admissibility while showing that admissibility equivalence does not determine degree. The source presents this as a conjecture and gives no evidence of resolution.