Fathi–Nelson free transport conjecture

Let $X=(X_1,\ldots,X_n)$ be a noncommutative random vector with joint law $\mu_X$, let $V_0$ and $V_1$ be potentials, and let $A=A^{\dagger}$ be a free Stein kernel for $X$ with respect to $V_1$. The quantity $\Sigma^*(X|V_0)$ denotes the relative free Stein discrepancy, and $L(\mathbb{F}_n)$ is the free group factor. Fathi–Nelson conjecture. There exists $\epsilon>0$ such that, if $\Sigma^*(X|V_0)<\epsilon$, then there exists a free transport from $\mu_X$ to the semicircular law; in particular,

$$W^*(X)\hookrightarrow L(\mathbb{F}_n).$$

The conjecture proposes that sufficiently small free Stein discrepancy yields free transport and an embedding into the free group factor. The source presents it as an open conjecture attributed to Fathi and Nelson.