The primitive-number square-free and order conjecture

Let $m$ be a primitive number, and write its prime decomposition as $m=p_1\dots p_r$. For a prime $p$, let $o_4(p)$ denote the order associated with $p$ in the paper.

Primitive-number conjecture. The number $m$ is square-free, and there exists an index $i$ such that

$$\operatorname*{\lcm}(o_4(p_1),\dots,o_4(p_r))=o_4(p_i).$$

This conjecture is motivated by the primitive numbers listed in Table 1 and gives structural restrictions on such numbers. The supplied text does not state whether it has been proved or disproved.