In 1989 Helge Tverberg proposed a quite general conjecture in Discrete Geometry, which could be considered as the common basis for many results in Combinatorial Geometry, and at the same time as a discrete analogue of the common transversal theorems. It implies or contains as special cases many classical "coincidence" results such as Radon's theorem, Rado's theorem, the Ham sandwich theorem, "non-embeddability" results (e.g. non-embeddability of graphs K 5 and K 3,3 in R 2 ), etc. The main goal of this short note is to verify this conjecture in one new, non-trivial case. We obtain the continuous version of the conjecture. So, it is not surprising that we use topological methods, or more precisely the methods of equivariant topology and the theory of characteristic classes.