First, you need to forget about what used to happen under UHS. It's irrelevant.
The rules (5.4) state the following:
"A player’s Handicap Index should be updated no later than the day after a score was submitted, or as soon as possible thereafter.
In a situation when a new round is played before the player’s Handicap Index has been updated, including when multiple rounds are played on the same day, it is recommended that the player uses their existing Handicap Index. However, in certain circumstances, the Committee in charge of the competition (or the Handicap Committee) has the discretion to decide what Playing Handicap the player should use (see Rule 7.2)."
CONGU's guidance (G5.4) states the following:
"When a player plays in more than one competition on a single day, or in a 36-hole competition on one day, his/her handicap will not be automatically re-calculated between rounds."