The exact-period conjecture for eventual quasilinearity of the omega-function

Let $\Gamma$ be a numerical monoid with least generator $n_1$, and let $\omega(n)$ denote its omega-function. Exact-period conjecture. For all sufficiently large $n$, the function $\omega(n)$ is quasilinear with period exactly $n_1$.

Eventual quasilinearity is known, with the period guaranteed to divide $n_1$; computational data in the source consistently gives period exactly $n_1$. The conjecture asks whether this exact period always occurs.