Dominant Alice II

Bob proposes a game to Alice: Each of them are given a coin that they are allowed to decide the probability of heads for. Let $p_1$ and $p_2$ be the probabilities that Alice and Bob select, respectively. They both flip their coins. If both coins come up heads, Alice gives Bob $€6$. If both come up tails, Alice gives Bob $€4$. Otherwise, if there is one head and one tail, Bob gives Alice $€5$. Find the largest value of $p_2$ that Bob can select such that he still has non-positive expected value when $p_1 = 7/9$.