Abelianness conjecture for commutators of normal subgroups with two minimal invariant character degrees

Let $N$ be a normal subgroup of the group $G$. The set of minimal $G$-invariant character degrees of $N$ is

$$\Mcd G N = \{\widehat{\theta}(1)\mid \theta \in \operatorname{Irr}(N)\}.$$

Commutator abelianness conjecture. If $\Mcd G N=\{1,f\}$ for some positive integer $f$, then the commutator subgroup

$$[N,G]$$

is abelian.

The paper notes that $N'$ is known to be abelian under these hypotheses, but conjectures that the stronger conclusion $[N,G]$ is abelian also holds.