Normally $\zeta$-reversible profinite groups

We examine (finitely generated) profinite groups in which two formal Dirichlet series, the normal subgroup zeta function and the normal probabilistic zeta function, coincide; we call these groups normally $\zeta$-reversible. We conjecture that these groups are pronilpotent and we prove this conjecture if $G$ is a normally $\zeta$-reversible satisfying one of the following properties: $G$ is prosoluble, $G$ is perfect, all the nonabelian composition factors of $G$ are alternating groups.