Unimodality conjecture for symplectically rational 8-manifolds

Let $(M,\omega)$ be a closed symplectically rational $8$-manifold admitting a Hamiltonian $S^1$-action. The even Betti numbers of $M$ are unimodal, meaning that they are weakly increasing up to the middle degree.

Unimodality conjecture. If $(M,\omega)$ is a closed symplectically rational $8$-manifold admitting a Hamiltonian $S^1$-action, then the even Betti numbers of $M$ are unimodal.

This conjecture extends known unimodality results for certain $8$-dimensional closed symplectic manifolds with Hamiltonian $S^1$-actions. Its status is not specified in the source.