The barycenter conjecture for one-dimensional free Gibbs measures

Let $f:\mathbb{R}\rightarrow\textbackslash \mathbb{R}$ be continuous and satisfy the assumptions denoted by $(\mathrm{gibbs})$. For $\lambda\textbackslash in\mathbb{R}$, let the free Gibbs measure associated with $f+\lambda\,\operatorname{id}$ be the corresponding equilibrium measure, and call its barycenter $\int x\,d\nu_{f+\lambda\,\operatorname{id}}(x)$. Barycenter conjecture. There exists $\lambda\in\mathbb{R}$ such that the free Gibbs measure associated with $f+\lambda\,\operatorname{id}$ has barycenter zero, that is,

$$\int x\,d\nu_{f+\lambda\,\operatorname{id}}(x)=0.$$

Establishing this would remove the main obstacle to the centered reduction used in the proof of the sharp symmetrized free transport-entropy inequality; the supplied text presents it as an unresolved question.