Permissive strategies for reachability games

A strategy in a two player game on graphs is said to be "permissive" if it is winning and it allows all behaviours of all winning memoryless strategies. For safety objectives, there is a most permissive strategy, and this most permissive strategy is in fact memoryless [Bernet, Janin, Walukiewicz'2002]. On the other hand, for reachability objectives, permissive strategies require memory. What is the complexity of the following problem: Given a reachability game and a constant c, does there exist a permissive strategy with memory at most c? It can be assumed that the constant c is at most the size of the game. So its encoding does not matter.