  We consider various statements that characterize the hyperfinite II$_1$
factors amongst embeddable II$_1$ factors in the non-embeddable situation. In
particular, we show that "generically" a II$_1$ factor has the Jung property
(which states that every embedding of itself into its ultrapower is unitarily
conjugate to the diagonal embedding) if and only if it is self-tracially stable
(which says that every such embedding has an approximate lifting). We prove
that the enforceable factor, should it exist, has these equivalent properties.
Our techniques are model-theoretic in nature. We also show how these techniques
can be used to give new proofs that the hyperfinite II$_1$ factor has the
aforementioned properties.
