Mañé's weak conjecture on generic Mather sets

Let $M$ be a manifold and let $L$ be a Tonelli Lagrangian on $TM$. For a cohomology class $c\in H^1(M,\mathbb{R})$, write $\mathcal{M}(L,c)$ for the Mather set associated with $L$ and $c$. Mañé's weak conjecture. There exists a residual subset $\mathcal{O}_2(L)\subset C^{\infty}(M)$ such that, for every $f\in\mathcal{O}_2(L)$, there is an open dense subset $U(L,f)\subset H^1(M,\mathbb{R})$ for which, for every $c\in U(L,f)$, the Mather set of $(L,c)$ consists of one periodic orbit. This is described as weaker than the strong conjecture because it concerns an open dense set of cohomology classes in addition to potential perturbations; the paper proves that the strong conjecture implies it, but the conjecture itself is presented as open.