It has been recently conjectured that for large systems, the shape of the central part of the large deviation function of the growth velocity would be universal for all the growth systems described by the Kardar-Parisi-Zhang equation in 1+1 dimension. One signature of this universality would be that the ratio of cumulants R(t)=[<h(3)(t)>(c)](2)/[<h(2)(t)>(c)<h(4)(t)>(c)] would tend towards a universal value 0.415 17ellipsis as t tends to infinity, provided periodic boundary conditions are used. This has recently been questioned by Stauffer. In this paper we summarize various numerical and analytical results supporting this conjecture, and report in particular some numerical measurements of the ratio R(t) for the Eden model.