Width-two conjecture for prime-indexed generalised Fabrykowski-Gupta groups

Let $p$ be an odd prime, and let $\Gamma_p$ be the corresponding generalised Fabrykowski-Gupta group. The width-two conjecture. Then $\Gamma_p$ is a group of width $2$. The conjecture is motivated by computed lower central series factors for $p=5,7,11$ and by the known pattern for $\Gamma_3$, but the source gives no proof or resolution.