Commutative minimal semigroup actions with non-semiopen maps

Determine whether there exists a commutative semigroup acting minimally on a compact space by continuous surjections, with at least one acting map that is not semiopen.

Background

The paper explains that, unlike group actions, minimal semigroup actions need not be surjective or semiopen. It gives a non-commutative example of a minimal action by continuous surjections that is not semiopen, and then identifies the commutative case as unresolved.

This problem concerns the extent to which the regularity properties of minimal actions depend on commutativity. Resolving it would clarify whether commutativity imposes additional topological structure on minimal semigroup actions.