  Let $(M,g)$ be a smooth compact Riemannian manifold of dimension $n$ with
smooth boundary $\partial M$, admitting a scalar-flat conformal metric. We
prove that the supremum of the isoperimetric ratio over the scalar-flat
conformal class is strictly larger than the best constant of the isoperimetric
inequality in the Euclidean space, and consequently is achieved, if either (i)
$9\le n\le 11$ and $\partial M$ has a nonumbilic point; or (ii) $7\le n\le 9$,
$\partial M$ is umbilic and the Weyl tensor does not vanish identically on the
boundary. This is a continuation of the work \cite{Jin-Xiong} by the second
named author and Xiong.
