Sharp upper bound for a branched transport problem coming from Ginzburg-Landau models

We consider a branched transport type problem with weakly imposed boundary conditions, which can be seen as a blown-up version of a reduced model for type-I superconductors in the regime of vanishing external magnetic field. We prove that if the irrigated measure is (locally) Ahlfors regular then it is of dimension at most $8\mathrm{/}5$ in agreement with the conjecture by Conti, the third author and Serfaty.