  We prove a new theorem of Tverberg type which confirms the conjecture of
Blagojevic, Frick, and Ziegler about the existence of "balanced Tverberg
partitions" (Conjecture 6.6 in, Tverberg plus constraints, Bull. London Math.
Soc., 46 (2014) 953-967). The proof relies on the connectivity and shellability
properties of multiple chessboard complexes and their symmetric analogues.
