The gcd conjecture for cyclotomic factors of a divisibility sequence

Let $(a_n)$ be the divisibility sequence with factorization $a_n=\prod_{d\mid n}b_d$, and let $\rho(p)$ denote the least positive index such that $p\mid a_{\rho(p)}$. For indices $m,n$, write $(b_m,b_n)$ for their greatest common divisor.

Gcd conjecture. If $(b_m,b_n)>1$, then

$$\frac{m}{n}=p^\alpha$$

\nfor some prime $p$ and some exponent $\alpha$.

This conjecture describes when two distinct factors in the divisor factorization of a divisibility sequence can share a prime divisor. The supplied text does not state whether it has been proved or disproved.