HMMT 2011 T7

Let $T$ denote the number of coins determined tails and $H$ denote the number of coins determined heads at the end of the game. Let $p_{m}(k)$ be the probability that $|T-H|=k$ after a game with $4 m$ coins, assuming that the player follows the optimal strategy outlined in problem 6. Clearly $p_{m}(k)=0$ if $k$ is an odd integer, so we need only consider the case when $k$ is an even integer. (By definition, $p_{0}(0)=1$ and $p_{0}(k)=0$ for $\left.k \geq 1\right)$. Prove that $p_{m}(0) \geq p_{m+1}(0)$ for all nonnegative integers $m$.