  We prove a general multidimensional invariance principle for a family of
U-statistics based on freely independent non-commutative random variables of
the type $U_n(S)$, where $U_n(x)$ is the $n$-th Chebyshev polynomial and $S$ is
a standard semicircular element on a fixed $W^{\ast}$-probability space. As a
consequence, we deduce that homogeneous sums based on random variables of this
type are universal with respect to both semicircular and free Poisson
approximations.
  Our results are stated in a general multidimensional setting and can be seen
as a genuine extension of some recent findings by Deya and Nourdin; our
techniques are based on the combination of the free Lindeberg method and the
Fourth moment Theorem.
