The paper is concerned with the limit shape (under some probability measure) of convex polygonal lines on Z+ starting at the origin and with the right endpoint n = (n1, n2) → ∞. In the case of the uniform measure, the explicit limit shape γ∗ was found independently by Vershik, Bárány and Sinai. Bogachev and Zarbaliev recently showed that the limit shape γ∗ is universal in a certain class of measures analogous to multisets in the theory of decomposable combinatorial structures. In the present work, we extend the universality result to a much wider class of measures, including (but not limited to) analogues of multisets, selections and assemblies. This result is in sharp contrast with the one-dimensional case, where the limit shape of Young diagrams associated with integer partitions heavily depends on the distributional type.