The closure conjecture for the first omitting cardinal of Magidority

Let $\lambda$ be Magidor and let $\alpha=\alpha_M(\lambda)$.

Closure conjecture. If $\theta<\alpha$, then

$$\theta^\omega<\alpha.$$

In particular, $\alpha_M(\lambda)$ cannot be $\mu^+$ when $\mu>\operatorname{cf}(\mu)=\omega$.

The conjecture would give a ZFC limitation on the possible value of the first omitting cardinal for Magidority, ruling out successors of singular cardinals of countable cofinality. The supplied text presents this as the authors' belief; no resolution is stated.