Occurrence of the product cohomology type for 3-torus actions

Determine whether a fixed-point component of a positively curved manifold with an effective isometric 3-torus action can have rational cohomology ring isomorphic to that of S^2 × HP^k.

Background

Theorem B classifies the rational cohomology ring of every fixed-point component of an oriented closed positively curved manifold admitting an effective isometric T^3-action. The possible rings are those of a sphere, complex projective space, quaternionic projective space, or S^2 × HP^k.

The paper explains that the S^2 × HP^k case is excluded for actions by a 4-torus or 5-torus, while all the other listed cases are known to occur. Whether this exceptional product type can occur under the weaker 3-torus symmetry assumption remains unresolved.