Mae9's generic uniqueness conjecture for minimizing measures

Let $M$ be a closed manifold, let $L$ be a generic Lagrangian on $M$, and let $H^1(M,\mathbb{R})$ be the first real cohomology group. There exists a dense open subset $U_0\subset H^1(M,\mathbb{R})$ such that, for every $\omega\in U_0$, the set $\mathcal{M}_{\omega}(L)$ of minimizing measures consists of a single periodic orbit or a fixed point.

Ma~n'e's conjecture. For a generic Lagrangian $L$ on a closed manifold $M$, there exist a dense open set $U_0$ of $H^1(M,\mathbb{R})$ such that for every $\omega\in U_0$, $\mathcal{M}_{\omega}(L)$ consists of a single periodic orbit or fixed point.

This is a genericity conjecture concerning the structure and uniqueness of minimizing measures in Lagrangian dynamics. The supplied text does not state whether it has been resolved, so its status is recorded as open.