The order conjecture for the symbol $[0,0,1]$

For each integer $N\ge 2$, consider the symbol $[0,0,1]$ in the module $\mathcal B_3(\mathbb Z/N\mathbb Z)$. The order conjecture. The element has order $1$, equivalently $[0,0,1]=0$, when $N$ is composite or $N\in\{2,3,5\}$, and has order exactly

$$\frac{p^2-1}{24}$$

when $N=p\ge 7$ is prime. Computer experiments for $N\le 23$ are stated to support this conjecture.