Mañé's strong conjecture on generic minimizing measures

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

$$\mathcal{O}_1(L)=\{f\in C^{\infty}(M):\text{the Mather set of }L+f\text{ consists of one periodic orbit}\}.$$

Mañé's strong conjecture. The set $\mathcal{O}_1(L)$ is residual in $C^{\infty}(M)$. Equivalently, for a generic potential perturbation, there is a unique minimizing measure and it is supported on a periodic orbit. The paper notes that an analogous conjecture can be formulated with $C^k(M)$ for any $k\geq2$; the conjecture is presented as open.