De Giorgi's reverse approximation conjecture for gradient flows

Let $\mathbb H$ be a finite-dimensional Euclidean space, and let $\phi:\mathbb H\to\mathbb R$ be a continuously differentiable Lipschitz function. A map $u\in C^1([0,\infty);\mathbb H)$ is a solution of the gradient-flow equation if and only if there exists, for every $\tau>0$, a Lipschitz perturbation $\phi_\tau:\mathbb H\to\mathbb R$ of $\phi$ such that

$$\lim_{\tau\downarrow 0}\operatorname{Lip}[\phi_\tau-\phi]=0,$$

and, for the generating functional

$$\Phi(\tau,U,V):=\frac{1}{2\tau}|V-U|^2+\phi_\tau(V),$$

one has $u\in\operatorname{GMM}(\Phi,u(0))$. De Giorgi's reverse approximation conjecture. Under these hypotheses, the stated equivalence holds: every solution is generated by minimizing movements for a family of perturbations converging to $\phi$ in Lipschitz seminorm, and every such minimizing movement is a solution. This conjecture asks whether all solutions of a gradient flow driven by a continuously differentiable Lipschitz function can be obtained through this restrictive reverse-approximation scheme; it is attributed to De Giorgi and is presented here without evidence of resolution.