  Let $(X_1,\dots,X_m)$ be self-adjoint non-commutative random variables
distributed according to the free Gibbs law given by a sufficiently regular
convex and semi-concave potential $V$, and let $(S_1,\dots,S_m)$ be a free
semicircular family. We show that conditional expectations and conditional
non-microstates free entropy given $X_1$, \dots, $X_k$ arise as the large $N$
limit of the corresponding conditional expectations and entropy for the random
matrix models associated to $V$. Then by studying conditional transport of
measure for the matrix models, we construct an isomorphism
$\mathrm{W}^*(X_1,\dots,X_m) \to \mathrm{W}^*(S_1,\dots,S_m)$ which maps
$\mathrm{W}^*(X_1,\dots,X_k)$ to $\mathrm{W}^*(S_1,\dots,S_k)$ for each $k = 1,
\dots, m$, and which also witnesses the Talagrand inequality for the law of
$(X_1,\dots,X_m)$ relative to the law of $(S_1,\dots,S_m)$.
