Uniqueness conjecture for invariant Ricci solitons on the four-sphere

Let $(g,u)$ be an $SO(2)\times SO(3)$-invariant solution of the gradient Ricci soliton equation on $\mathbb{S}^4$ with soliton constant $\lambda=1$. The equation is understood with respect to the metric $g$ and potential function $u$.

Uniqueness conjecture. Any such solution is the round sphere, up to diffeomorphism.

The conjecture seeks to determine uniqueness among invariant shrinking Ricci solitons on $\mathbb{S}^4$. The paper proves a compactness result for these solutions, including uniform injectivity-radius, volume, and pointwise curvature bounds, but the stated uniqueness remains open.