Nearly a month later and the swarm is still there, although its size seems diminished.
That sentence leads me to believe that the bees probably have no queen & will perish.
If a queen stops laying or dies & there are no very fresh eggs for the workers to make a new queen sometimes the bees leave & swarm as you describe. Unless they can be added to an existing colony with a queen they will gradually die off. When they swarm
with a queen they usually find a new home quite fast.