Abdollahi–Akbari–Maimani nilpotency conjecture for non-commuting graphs

Let $G$ be a finite non-abelian nilpotent group, and let $H$ be a group. For a finite non-abelian group $K$, let $\Gamma_K$ be the graph whose vertices are the non-central elements of $K$, with two vertices adjacent when they do not commute. Abdollahi–Akbari–Maimani nilpotency conjecture. If

$$\Gamma_G\cong\Gamma_H,$$

then $H$ is nilpotent. This conjecture asks whether nilpotency can be recovered from the non-commuting graph. The source states that it remains open, although it is proved for finite AC-groups when $|Z(G)|\geq|Z(H)|$ and substantial partial results are known.