Summary: We prove that weak solutions of systems with skew-symmetric structure, which possess a continuous boundary trace, have to be continuous up to the boundary. This applies, e.g., to the \(H\)-surface system \(\Delta u= 2H(u)\partial_{x^1}u\wedge \partial_{x^2}u\) with bounded \(H\) and thus extends an earlier result by P. Strzelecki and proves the natural counterpart of a conjecture by E. Heinz. Methodically, we use estimates below natural exponents of integrability and a recent decomposition result by T. Rivière.