I would consider June's sense of willpower to be, how do I put this...
Strong, but divided.
There's no debating that a part of her's left compromised by her dark and mysterious past, particularly in terms of motivation, but god DAMN if she doesn't have grit and focus alike regardless. No matter how much she doesn't like herself or her past, she's still a swordmaster, and that sticks with her no matter what. I mean, imagine taking a sledgehammer to the dome and keeping it up.