BCP for surfaces with rational polyhedral Mori cones

Determine whether every smooth projective surface with a rational polyhedral Mori cone satisfies the bounded cohomology property.

Background

A rational polyhedral Mori cone is generated by finitely many curve classes. The paper explains that the bounded cohomology property has been established for several special classes, including Mori dream surfaces, but that rational polyhedrality alone may not ensure the required cohomological bound.

The question asks whether rational polyhedrality of the Mori cone is sufficient for the bounded cohomology property on every smooth projective surface.