Made's weak genericity conjecture for minimizing measures

Let $M$ be a closed manifold, let $L$ be a Tonelli Lagrangian on $M$, and let $H^1(M,\mathbb R)$ denote the first cohomology group. For a generic Lagrangian $L$, consider $(L,c)$-minimizing measures, obtained after perturbing by a closed $1$-form representing $c$.

Made's weak conjecture. For a generic Lagrangian $L$, there is an open dense subset $U$ of $H^1(M,\mathbb R)$ such that, for every $c\in U$, there is a unique $(L,c)$-minimizing measure, and this measure is supported on a periodic orbit or a fixed point of the Euler–Lagrange flow.

The weak version allows closed $1$-form perturbations in addition to potentials and is presented as part of Made's program on generic minimizing measures. The supplied text does not state whether it has been resolved.