Classification conjecture for compact simply-connected 4-dimensional Ricci solitons

A compact simply-connected $4$-dimensional Ricci soliton is a compact Riemannian $4$-manifold with a gradient Ricci soliton structure. An action is cohomogeneity one if its generic orbits have codimension one.

Classification conjecture. If a compact simply-connected $4$-dimensional Ricci soliton admits an invariant cohomogeneity one action, then it is isometric to a rescaling of one of the following:

• the Koiso--Cao metric;
• the Page metric;
• the Fubini--Study metric on $\mathbb{CP}^2$;
• the round metric on $\mathbb{S}^4$; or
• the direct product of round metrics on $\mathbb{S}^2 \times \mathbb{S}^2$ with the same radii.

Only a small number of compact cohomogeneity-one constructions are known in this setting, including the Koiso--Cao and Page metrics and the standard homogeneous examples. The conjecture proposes that this list exhausts the compact simply-connected four-dimensional Ricci solitons admitting an invariant cohomogeneity-one action.