Bardakov–Nasybullov–Neshchadim's nilpotency conjecture for groups with subgroup twisted conjugacy classes

Let $G$ be a group, let $\operatorname{Aut}(G)$ denote its automorphism group, and for $\varphi\in\operatorname{Aut}(G)$ let $[e]_{\varphi}$ be the twisted conjugacy class of the identity element $e$ under $\varphi$. Bardakov–Nasybullov–Neshchadim's conjecture. If $[e]_{\varphi}$ is a subgroup of $G$ for every $\varphi\in\operatorname{Aut}(G)$, then $G$ is nilpotent. The conjecture asks whether this condition on twisted conjugacy classes forces nilpotency; the supplied source gives no evidence of a resolution.