The order-determination conjecture for non-commuting graphs

Let $G$ and $H$ be finite non-abelian groups. For a group $G$, let $\Gamma_G$ denote its non-commuting graph, whose vertices are the elements of $G\setminus Z(G)$, with two distinct vertices adjacent exactly when they do not commute.

Order-determination conjecture. If

$$\Gamma_G\cong \Gamma_H,$$

then

$$|G|=|H|.$$

The conjecture was refuted by exhibiting finite groups $G$ and $H$ with isomorphic non-commuting graphs but different orders, specifically $|G|=2^{10}\cdot 5^3$ and $|H|=2^3\cdot 5^6$.