There are other mechanisms that tend to push it toward $1 such as people choosing 50/50 HBD/HP or 100% HP for rewards, and speculation pushing it away from the bounds (makes more sense to sell HBD at 1.05 than to buy it, and likewise buy at 0.95 than sell).
I really hope you're right. I think it would tend to stay in the lower range of $0.95 more often than not as buying HBD at $1 would mean taking a certain time commitment to make up for the 5% fee. Around 3 months before ROI if the savings rate was 20% like currently.
Only converting would incur the fee. If you buy from the market you're not paying a fee. Likewise selling. If you buy from the market at $1 you have essentially symmetric risk, since you can lose 5% if it drops to 0.95 but you can also gain 5% if it rises to 1.05. Neither of these would inherently dominate.