The converse nilpotence conjecture for supercharacter theories

Let $G$ be a finite group and let $\mathsf{S}$ be a supercharacter theory of $G$. For each $\chi\in\mathrm{Ch}(\mathsf{S})$, let $\ker(\chi)$ denote its kernel.

Converse nilpotence conjecture. If

$$\chi(1)\mid\lvert G:\ker(\chi)\rvert$$

for all $\chi\in\mathrm{Ch}(\mathsf{S})$, then $G$ is $\mathsf{S}$-nilpotent. In particular, $G$ is nilpotent.

For nilpotent groups, the analogous divisibility condition with $\chi(1)^2$ and ordinary irreducible characters is known, and its converse was proved by Gagola and Lewis. The stated supercharacter-theoretic generalization is motivated by computational evidence and remains open in the supplied source.