We had this question the other day because my opponent started the game with 2x of the DS objective that reduces your reserve by 1 while undamaged. On my first turn, would I be forced to discard down to 4 during my draw phase?
Are you forced to discard down to your reserve value?
Yes.
"After he has discarded a card, or chosen not to, the
active player must return his hand to a number of cards
equal to his current reserve value, drawing from his deck
or choosing and discarding from his hand as necessary."
Yes, you are. Check page 12 of the rulebook under Draw Phase.