Some remarks on Kida's formula when $\mu\neq 0$

The Kida's formula in classical Iwasawa theory relates the Iwasawa $\lambda$-invariants of $p$-extensions of number fields. Analogue of this formula was subsequently established for the Iwasawa $\lambda$-invariants of Selmer groups under an appropriate $\mu=0$ assumption. In this paper, we give a conceptual (but conjectural) explanation that such a formula should also hold when $\mu\neq 0$. The conjectural component comes from the so-called $\mathfrak{M}_H(G)$-conjecture in noncommutative Iwasawa theory.