The coprime-order criterion for completeness

Let $m$ be an odd number not divisible by $3$, with prime decomposition

$$m=p_1^{k_1}\dots p_r^{k_r}.$$

Assume that the numbers $o_4(p_1),\dots,o_4(p_r),p_1,\dots,p_r$ are mutually prime.

Coprime-order completeness conjecture. Then $m$ is complete.

This is explicitly presented as a weaker conjecture than the preceding primitive-number conjecture. The supplied text does not state whether it has been proved or disproved.