Associativity conjecture for finite Moufang p-loops with p not equal to 3

Let $L$ be a finite Moufang $p$-loop, where $p\ne 3$. Suppose $L$ is generated by a set $\{a_i\mid i\in I\}$ satisfying

$$(a_i,a_j,a_k)=1$$

for all $i,j,k\in I$, where $(x,y,z)$ denotes the associator defined by $(xy)z=(x(yz))(x,y,z)$. Associativity conjecture. Then $L$ is associative. The conjecture excludes the prime $3$, which is essential in the paper's constructed counterexample of order $3^{19}$; the stated result is presented as a conjectural affirmative answer to the preceding problem for finite Moufang $p$-loops with $p\ne 3$.