Eh... doubtful, but maybe. It's really hard to say. Carr hasn't really been bad even in the losses (he was ranked as the top-performing QB going into this game, I believe)... but the offense has been sputtering.
If Rattler wins both of these next 2 games, and he does so putting up a lot of points and with the offense getting back to the efficiency we saw early in the season, the Saints might have a decision to make. But short of that, I don't think Carr's starting job is in jeopardy.
I think Rattler would have be leading the team to wins in overwhelming fashion, putting up 30+ points on offense in each game for it to even be a conversation.