We consider a branched transport problem with weakly imposed boundary conditions. This problem arises as a reduced model for pattern formation in type-I superconductors. For this model, it is conjectured that the dimension of the boundary measure is non-integer. We prove this conjecture in a simplified 2D setting, under the (strong) assumption of Ahlfors regularity of the irrigated measure. This work is the first rigorous proof of a singular behaviour for irrigated measures resulting from minimality.