  The $C^{\ast}$-algebra $\mathcal{U}_{nc}(n)$ is the universal
$C^{\ast}$-algebra generated by $n^2$ generators $u_{ij}$ that make up a
unitary matrix. We prove that Kirchberg's formulation of Connes' embedding
problem has a positive answer if and only if $\mathcal{U}_{nc}(2)
\otimes_{\min} \mathcal{U}_{nc}(2)=\mathcal{U}_{nc}(2) \otimes_{\max}
\mathcal{U}_{nc}(2)$. Our results follow from properties of the
finite-dimensional operator system $\mathcal{V}_n$ spanned by $1$ and the
generators of $\mathcal{U}_{nc}(n)$. We show that $\mathcal{V}_n$ is an
operator system quotient of $M_{2n}$ and has the OSLLP. We obtain necessary and
sufficient conditions on $\mathcal{V}_n$ for there to be a positive answer to
Kirchberg's problem. Finally, in analogy with recent results of Ozawa, we show
that a form of Tsirelson's problem related to $\mathcal{V}_n$ is equivalent to
Connes' Embedding problem.
