Coin splitting game

Just a fun question I've been thinking about: The following zero-sum game is played between the Dealer and team Alice & Bob. The Dealer receives an odd number N of fair coin tosses in sequence. Whenever a coin is tossed, the Dealer sees the result and then chooses whether to show it to Alice or Bob. Both players are aware of the turns on which they receive the coin flip results. At the end of the N tosses, both Alice and Bob simultaneously guess whether there are more heads or tails in total, and score one point per correct guess. We allow all players including the Dealer to randomize their choices. Alice and Bob can agree on a strategy beforehand, possibly involving some initial coordinated randomization but cannot communicate after the game starts. Team Alice & Bob can score at least an expected payoff of 1 by guessing randomly and independently. The question is thus how much they can improve on this. Question: Let V_N be the value of the N coin game for team Alice & Bob. Is it true that V_N = 1 +KN^{-1/2} + o(N^{-1/2})? for some constant K > 0? If so, to what extent can we determine K and the corresponding optimal, or asymptotically optimal strategies?