  We study upgraded free independence phenomena for unitary elements $u_1$,
$u_2$, \dots representing the large-$n$ limit of Haar random unitaries, showing
that free independence extends to several larger algebras containing $u_j$ in
the ultraproduct of matrices $\prod_{n \to \mathcal{U}} M_n(\mathbb{C})$. Using
a uniform asymptotic freeness argument and volumetric analysis, we prove free
independence of the Pinsker algebras $\mathcal{P}_j$ containing $u_j$. The
Pinsker algebra $\mathcal{P}_j$ is the maximal subalgebra containing $u_j$ with
vanishing $1$-bounded entropy defined by Hayes; $\mathcal{P}_j$ in particular
contains the relative commutant $\{u_j\}' \cap \prod_{n \to \mathcal{U}}
M_n(\mathbb{C})$, more generally any unitary that can be connected to $u_j$ by
a sequence of commuting pairs of Haar unitaries, and any unitary $v$ such that
$v\mathcal{P}_j v^* \cap \mathcal{P}_j$ is diffuse. Through an embedding
argument, we go back and deduce analogous free independence results for
$\mathcal{M}^{\mathcal{U}}$ when $\mathcal{M}$ is a free product of Connes
embeddable tracial von Neumann algebras $\mathcal{M}_i$, which thus yields (in
the Connes-embeddable case) a generalization and a new proof of
Houdayer--Ioana's results on free independence of approximate commutants. It
also yields a new proof of the general absorption results for Connes-embeddable
free products obtained by the first author, Hayes, Nelson, and Sinclair.
