In superconformal field theories in four space–time dimensions, the quantum corrections with four derivatives are believed to be severely constrained by non-renormalization theorems. The strongest of these is the conjecture formulated by Dine and Seiberg in hep-th/9705057 that such terms are generated only at one loop. In this note, using the background field formulation in superspace, we test the Dine–Seiberg proposal by comparing the two-loop F4 quantum corrections in two different superconformal theories with the same gauge group SU(N): (i) SYM (i.e., SYM with a single adjoint hypermultiplet); (ii) SYM with 2N hypermultiplets in the fundamental. According to the Dine–Seiberg conjecture, these theories should yield identical two-loop F4 contributions from all the supergraphs involving quantum hypermultiplets, since the pure SYM and ghost sectors are identical provided the same gauge conditions are chosen. We explicitly evaluate the relevant two-loop supergraphs and observe that the F4 corrections generated have different large N behaviour in the two theories under consideration. Our results are in conflict with the Dine–Seiberg conjecture.