Remove starting window on the corresponding task organizer.

There can be multiple task organizer(s) register to core and exist at
the same time, to ensure the starting window can removed at the correct
client without mass up the starting window lifecycle, record the
organizer which create the starting window, so it can be removed on it
correctly.

Bug: 295621224
Test: presubmit
Change-Id: I207a25f6c8280a32e90b475902e2150929103192