Mañé's weak Aubry-set conjecture

Let $M$ be a manifold and let $L$ be a Tonelli Lagrangian on $TM$. For a cohomology class $c\in H^1(M,\mathbb{R})$, let the Aubry set of $(L+f,c)$ denote the Aubry set associated with the potential perturbation $L+f$ and class $c$. Mañé's weak Aubry-set conjecture. There exists a residual subset $\mathcal{O}_4(L)\subset C^{\infty}(M)$ 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 Aubry set of $(L+f,c)$ consists of one hyperbolic periodic orbit. The source presents this as a consequence of the Aubry-set conjecture and notes that it implies the weak Mather-set conjecture; the statement is not established in the paper and is treated as open.