Petrovic's 4-king conjecture for multipartite hypertournaments

Let $H$ be a multipartite $k$-hypertournament, where $k\ge 2$. A vertex is a transmitter if it has no incoming arc with any vertex from a different partite set, and a 4-king is a vertex from which every vertex is reachable by a path of length at most $4$. Petrovic's conjecture. If $H$ has at most one transmitter, then $H$ contains a 4-king. This generalizes the corresponding result for multipartite tournaments and was known for bipartite $k$-hypertournaments; the paper states that it will prove the conjecture affirmatively.