Answer
The torques about another choice of rotation axis will not balance.
Work Step by Step
In the Review & Summary section on page 343, the text states: "Static equilibrium also implies that the vector sum of the external torques acting on the body about any point is zero."
Suppose the torques about the rotation axis through point $O$ do not balance. Then we can conclude that the stick is not in equilibrium.
According to the text quoted above, if the torques about another choice of rotation axis balance, then the stick is in equilibrium. This is a contradiction.
Therefore, the torques about another choice of rotation axis will not balance.
