  We investigate the following optimization problem: what is the least possible
conformal capacity of a pair of linked curves in $S^3$? A natural conjecture,
due to Gehring, Martin and Palka, is that the optimal value is attained by the
standard Hopf link. We prove that this is the case under the assumption that
each component of the link lies on a different side of a conformal image of the
Clifford torus. In particular, this shows that the Hopf link is a local
minimizer in a strong sense.
