Hamilton's Yamabe flow convergence conjecture

Let $(X,g_0)$ be a closed Riemannian manifold of dimension $d\geq 3$, and let $g(t)$ be the unique solution of the Yamabe flow with initial metric $g_0$. A metric has constant scalar curvature when its scalar curvature is constant on $X$.

Hamilton's conjecture. The metric $g(t)$ converges to a metric of constant scalar curvature as $t\to\infty$.

Hamilton's conjecture concerns the long-time asymptotics of the Yamabe flow after global existence has been established. The source attributes the proposal to Hamilton and does not provide evidence of a resolution.