Bounded-diameter conjecture for commuting graphs of centerless finite groups

Let $G$ be a finite group with trivial centre, and let its commuting graph be the graph whose vertices are the non-central elements of $G$, with two distinct vertices adjacent exactly when they commute in $G$. Centerless commuting-graph conjecture. There is an absolute constant $d$ such that the commuting graph of $G$ is either disconnected or has diameter at most $d$. This is proposed after the unrestricted conjecture is disproved by finite special $2$-groups with commuting graphs of arbitrarily large diameter; the status of the centerless case is left open in the paper.