Asymptotical freeness conjecture for the polar part and modulus of a biinvariant ensemble

Let $(D_n)$ be a sequence of constant Hermitian matrices in ${\mathcal{M}}_n({\mathbb{C}})$ whose distributions converge and whose operator norms are uniformly bounded. Let $V_n$ be Haar-distributed in the unitary group $U(n)$, and set $x_n=V_nD_n$. The sequences $(\lim_{k\rightarrow\infty}\sqrt[2k]{(x_n^{\star})^k x_n^k})_{n\in{\mathbb{N}}}$ and $(x_nx_n^{\star})_{n\in{\mathbb{N}}}$ are asymptotically free, where asymptotic freeness is defined using normalized traces and the limiting free $\star$-algebraic distribution.

Asymptotical freeness conjecture. The sequences

$$\left(\lim_{k\rightarrow\infty}\sqrt[2k]{(x_n^{\star})^k x_n^k}\right)_{n\in{\mathbb{N}}}$$

and

$$(x_nx_n^{\star})_{n\in{\mathbb{N}}}$$

are asymptotically free.

The preceding results establish asymptotic freeness for each fixed $k$ and motivate this limiting assertion for the $U(n)$-biinvariant ensemble. The supplied text gives no resolution of the conjecture.