Suppose my strategy is to double at 2/3 + epsilon where epsilon < (4/5 - 2/3) instead of at your suggested value of 2/3.
Suppose you have just doubled at 2/3. Eventually the win chance will reach either 2/3 + epsilon or 1/2. In the former case we then are in the same position both having doubled and at the same win chance. In the second case we both have 1/2 win chance but I have cube access and you have given the cube to the opponent. I am in the better situation. Thus my strategy is better.
#35 on Giants on Backgammon list