Made's strong genericity conjecture for minimizing measures

Let $M$ be a closed manifold, let $L$ be a Tonelli Lagrangian on $M$, and let $L+f$ range over the generic perturbations by $C^2$ potentials described in the source. A minimizing measure is an invariant probability measure minimizing the action associated with the Lagrangian.

Made's strong conjecture. For a generic Lagrangian, there is a unique minimizing measure, and this measure is supported on a periodic orbit or a fixed point of the Euler–Lagrange flow.

This is one of Made's proposed genericity conjectures, intended to ensure that minimizing measures are sufficiently sparse for the construction of diffusing orbits. The supplied text does not state whether the conjecture has been resolved.