  We consider a branched transport type problem which describes the magnetic
flux through type-I superconductors in a regime of very weak applied fields. At
the boundary of the sample, deviation of the magnetization from being uniform
is penalized through a negative Sobolev norm. It was conjectured by S. Conti,
F. Otto and S. Serfaty that as a result, the trace of the magnetization on the
boundary should be a measure of Hausdorff dimension $8/5$. We prove that this
conjecture is equivalent to the proof of local energy bounds with an optimal
exponent. We then obtain local bounds which are however not optimal. These
yield improved lower bounds on the dimension of the irrigated measure but
unfortunately does not improve on the trivial upper bound. In order to
illustrate the dependence of this dimension on the choice of penalization, we
consider in the last part of the paper a toy model where the boundary energy is
given by a Wasserstein distance to Lebesgue. In this case minimizers are finite
graphs and thus the trace is atomic.
