  We define the notion of self-tracial stability for tracial von Neumann
algebras and show that a tracial von Neumann algebra satisfying the Connes
Embedding Problem is self-tracially stable if and only if it is amenable. We
then generalize a result of Jung by showing that a separable tracial von
Neumann algebra that satisfies the Connes Embedding Problem is amenable if and
only if any two embeddings into $R^\mathcal{U}$ are ucp-conjugate. Moreover we
show that for a II$_1$ factor $N$ satisfying CEP, the space $\mathbb{H}$om$(N,
\prod_{k\to \mathcal{U}}M_k)$ of unitary equivalence classes of embeddings is
separable if and only $N$ is hyperfinite. This resolves a question of Popa for
Connes embeddable factors. These results hold when we further ask that the
pairs of embeddings commute, admitting a nontrivial action of
$\text{Out}(N\otimes N)$ on $\mathbb{H}$om$(N\otimes N, \prod_{k\to
\mathcal{U}}M_k)$ whenever $N$ is non-amenable. We also obtain an analogous
result for commuting sofic representations of countable sofic groups.
