Mañé's Aubry-set conjecture for generic potential perturbations

Let $M$ be a closed manifold and let $L$ be an autonomous Tonelli Lagrangian on $TM$. Define

$$\mathcal{O}_3(L)=\{f\in C^{\infty}(M):\text{the Aubry set of }L+f\text{ consists of one hyperbolic periodic orbit}\}.$$

Mañé's Aubry-set conjecture. The set $\mathcal{O}_3(L)$ is residual in $C^{\infty}(M)$. The source states that this conjecture is equivalent to the strong conjecture on generic minimizing measures, using upper semicontinuity of the Aubry set when there is only one minimizing measure; it is presented as an apparently stronger, unresolved formulation.