The forbidden ordering conjecture for omega-values in 3-generated numerical monoids

Let $\Gamma = \langle n_1,n_2,n_3\rangle$ be a numerical monoid minimally generated by $n_1<n_2<n_3$, and let $\omega(n_i)$ denote the omega-value of $n_i$. The forbidden ordering conjecture. The following orderings are impossible:

$$\omega(n_1)>\omega(n_2)>\omega(n_3),\qquad \omega(n_1)=\omega(n_2)>\omega(n_3),\qquad \omega(n_3)<\omega(n_1)<\omega(n_2).$$

Computations indicate that almost every ordering of the three omega-values occurs for some minimally 3-generated numerical monoid, while these three orderings do not. The claim concerns the remaining obstruction to describing which orderings are possible.