Symmetric multiple chessboard complexes and a new theorem of Tverberg type

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.