  We show that any II$_1$ factor that has the same 4-quantifier theory as the
hyperfinite II$_1$ factor $\mathcal{R}$ satisfies the conclusion of the Popa
Factorial Commutant Embedding Problem (FCEP) and has the Brown property. These
results improve recent results proving the same conclusions under the stronger
assumption that the factor is actually elementarily equivalent to
$\mathcal{R}$. In the same spirit, we improve a recent result of the
first-named author, who showed that if (1) the amalgamated free product of
embeddable factors over a property (T) base is once again embeddable, and (2)
$\mathcal{R}$ is an infinitely generic embeddable factor, then the FCEP is true
of all property (T) factors. In this paper, it is shown that item (2) can be
weakened to assume that $\mathcal{R}$ has the same 3-quantifier theory as an
infinitely generic embeddable factor.
