uefa.com suggests otherwise. The Europa Conference draw structure implies top eight qualify then play-offs for 9th to 24th. That means 16 teams, and the next round is "round of 16" so there is nowhere to insert the CL cast-offs. This also applies in the Europa League.
It looks like teams losing the play-offs go out along with teams finishing 25th to 36th.
Teams losing qualifying games before the league stage drop into other competitions.
Please note: uefa.com doesn't specifically say those losing in the post-league play-offs go out, but there is nothing to suggest they drop to another tournament.