Square-freeness conjecture for the factors of a divisibility sequence

Let $(a_n)$ be a divisibility sequence with factorization $a_n=\prod_{d\mid n}b_d$. An integer is square-free if it is not divisible by the square of any prime.

Square-freeness conjecture. If $n$ is square-free, then $b_n$ is square-free.

The paper presents this as a stronger version of the open question of whether the relevant Fibonacci-type terms are square-free. The supplied status is unknown, so the conjecture is recorded as open.