It could drop at any time due to, for example, many chargers being connected to 1 circuit and the chargers coordinating with each-other to load-balance their connected vehicles (everyone gets their fair share), or it could be tied to solar PV generation so that charging doesn't import from the grid; the available current will change with how much excess PV is available, etc.
A well-engineered EVSE will notice when the EV is drawing more than the EVSE wants it to. It should then open the mains contactor at this point; cutting the EV's electricity supply. It will also hold the control pilot low to signal a fault to the EV.
Not saying this is the case here, of course, but they could do something similar that way if they wanted to...
If the AC "charger" attempted to regulate current by varying voltage the actual onboard charger would just attempt to pull more current to make up for the voltage drop and probably trip the whole thing.
There would be no point in regulating the AC current, it would make the charger very complicated and expensive replicating much of whats in the DC converter/charger in the car.
A full DC charger like at a charging station does do this because it is the actual charger bypassing the cars small internal DC charger with DC leads going directly to battery at full pack voltage.