Is the property management company collecting an extra commission each time a new tenant moves in? Because all this strategy is going to do is let tenants quit once the rent goes considerably above the market rate.

A much wiser strategy, IMO, would be to watch the market rates and keep the existing tenants' rate constant until the market rate climbs considerably higher (7% sounds like reasonable threshold) and then raising it to be marginally below the market, incentivizing the tenant to stay.