Akbari–Mohammadian–Radjavi–Raja's commuting graph diameter conjecture

Let $\mathbb{F}$ be a field and let $\operatorname{Mat}_n(\mathbb{F})$ be the algebra of $n\times n$ matrices over $\mathbb{F}$. Let $\Gamma(\mathbb{F},n)$ be the graph whose vertices are the non-scalar matrices in $\operatorname{Mat}_n(\mathbb{F})$, with an edge between $U$ and $V$ when $UV=VU$. Akbari–Mohammadian–Radjavi–Raja's conjecture. If $\Gamma(\mathbb{F},n)$ is connected, then its diameter does not exceed five. The conjecture concerns the maximal possible diameter of connected commuting graphs of matrix algebras. It is refuted: $\Gamma(\mathbb{F}_2,9)$ is connected and has diameter at least five, so the conjecture cannot be strengthened to a bound below five; however, the supplied evidence does not establish a counterexample to the stated upper bound of five.