Computational complexity of rank-4 Maker-Breaker games

Determine the computational complexity of deciding the winner of Maker-Breaker games played on hypergraphs of rank 4.

Background

The paper establishes that deciding the winner of Maker-Breaker games is PSPACE-complete for 5-uniform hypergraphs. It also notes that Maker-Breaker games on hypergraphs of rank 3 or lower are solvable in polynomial time. The complexity of the intervening rank-4 case is therefore left unresolved, forming the natural boundary problem between the known polynomial-time and PSPACE-complete results.