  We classify local minimizers of $\int\sigma_2+\oint H_2$ among all
conformally flat metrics in the Euclidean $(n+1)$-ball, $4\leq n\leq 5$, for
which the boundary has unit volume, subject to an ellipticity assumption. We
also classify local minimizers of the analogous functional in the critical
dimension $n+1=4$. If minimizers exist, this implies a fully nonlinear sharp
Sobolev trace inequality. Our proof is an adaptation of the Frank--Lieb proof
of the sharp Sobolev inequality, and in particular does not rely on
symmetrization or Obata-type arguments.
