Small free Stein discrepancy implies free transport to the semicircle law

Let $X=(x_1,\ldots,x_n)$ be an $n$-tuple with joint law $\mu_X$, and let $A=A^\dagger$ be a free Stein kernel for $X$ relative to the potential $V_1$. Write $\Sigma^*(X\mid V_1)$ for the free Stein discrepancy, and let $L(\mathbb{F}_n)$ denote the von Neumann algebra generated by a free $(0,1)$-semicircular $n$-tuple. Small free Stein discrepancy conjecture. There exists $\epsilon>0$ such that if

$$\Sigma^*(X\mid V_1)<\epsilon,$$

then there exists free transport from the free semicircle law to $\mu_X$. In particular, there is an embedding

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

The conjecture proposes that sufficiently small free Stein discrepancy is enough to trigger the free monotone transport mechanism, extending the known examples where free transport arguments identify the generated von Neumann algebra with a free group factor. The supplied text does not state whether this has been proved or disproved.