If the rents go up and a native couple that still remembers the living standards of their parents is forced to raise a family in two rooms, they will not have children. Immigrants used to that situation might.
There are other solutions, e.g. that the wealthy boomers pay with their houses for their retirements or are forced to rent out their huge properties.
The retirement ponzi scheme needs to stop at some point anyway. With automation one might also need fewer workers.
Most importantly, many immigrants receive social security and are not employed.