OEIS conjecture on moduli with one zero in the Pisano period

Let $m$ be a positive integer. The Pisano period modulo $m$ is one shortest period of the Fibonacci sequence modulo $m$, and its number of zeros is the order of $m$. OEIS conjecture A053031. An integer $m$ has one zero in its Pisano period if and only if $m$ is an odd number, all of whose factors have one zero in their Pisano period, or if $m$ is twice or four times such a number. This conjecture concerns the classification of moduli according to the number of zeros in their Pisano periods; the source presents it as an OEIS conjecture but gives no resolution.