Global existence and convergence conjecture for Yang–Mills gradient flow

Let $G$) be a compact Lie group and $P$ a principal $G$-bundle over a closed, connected, four-dimensional smooth manifold $X$ with Riemannian metric $g$. Let $A_0$ be a smooth connection on $P$. The Yang–Mills gradient flow is

$$\frac{\partial A}{\partial t}=-d_{A(t)}^{*_g}F_{A(t)},\qquad A(0)=A_0.$$

Global existence and convergence conjecture. There is a smooth solution $A(t)$ for $t\in[0,\infty)$ to this flow for the Yang–Mills energy functional with respect to the $L^2$ Riemannian metric on the affine space of connections on $P$. Moreover, as $t\to\infty$, $A(t)$ converges to a smooth Yang–Mills connection $A_\infty$ on $P$.

This conjecture is attributed in the source to Atiyah and Bott, Sedlacek, Taubes, and Uhlenbeck, and is stated as a global existence and asymptotic convergence problem for Yang–Mills flow on closed four-manifolds. The source does not provide evidence of a resolution.