Positive-curvature flow convergence conjecture

Let $M$ be a closed Riemannian manifold with positive sectional curvature. Consider the flow defined in the paper, denoted by

, and let $g$ be a Riemannian metric of positive sectional curvature. **Positive-curvature flow convergence conjecture.** For some initial vector field and a certain Riemannian metric $g$ of positive sectional curvature, the flow

converges uniformly to a nonzero Killing vector field with respect to $g$. The conjecture extends the convergence result proved in the Einstein case and, if true, would immediately answer Yau's long-standing question about the existence of an effective $\mathbb{S}^{1}$-action on a closed manifold with positive sectional curvature. The source does not state a resolution of this conjecture, although the parser marks the candidate as resolved; this status should be checked.