Nilpotent-factor conjecture for non-commuting graphs

Let $p$ be a prime, and suppose that $G=P\times A$ is a finite group, where $P\in\operatorname{Syl}_p(G)$ is non-abelian and $A$ is an abelian $p'$-group. Let $\Gamma_K$ denote the non-commuting graph of a finite non-abelian group $K$, whose vertices are the non-central elements and whose edges join pairs of noncommuting vertices. Nilpotent-factor conjecture. If $H$ is finite,

$$\Gamma_G\cong\Gamma_H\quad\text{and}\quad |Z(G)|\geq|Z(H)|,$$

then $H=Q\times B$, where $q$ is a prime, $Q\in\operatorname{Syl}_q(H)$ is non-abelian, and $B$ is an abelian $q'$-group. In particular, $H$ is nilpotent. This is proposed as a reduction for the unresolved nilpotency conjecture in the case where the nilpotent group has a unique non-abelian Sylow subgroup.