Yes, Autoquantize has been a feature for a while now, and what it does is exactly what you’re describing
The latest Aeros version can be found here:
The latest beta can be found here:
You can have each part autoquantize independently or just autoquantize the whole song’s tempo based on the very first track in part 1, this is all settable in the song settings. Set the Song Grid Mode to Auto to use autoquantize. You will also probably want to change the MIDI Out setting for the MIDI Start command to send on commit instead of at the beginning of the recording, this way any listening device won’t start playback until the Aeros has set the tempo.