I finished my first day of the adventure with one energy left over as I was at a dungeon that required two. The following day I had seven out of seven energy. The one left over from the previous day did not carry over.
This also happens on completion of the adventure. I had one energy left over but when I restarted the adventure it showed seven out of seven.
Purchased energy carries over to next day or next adventure start but “free” energy doesn’t. In my opinion, this makes sense since as it happens with season, PvP and event tokens it is a way to promote players to play everyday to maximize rewards, instead of just accumulating tokens/energy and doing everything on last day. However, it does leave a bad feeling when even after winning all fights you cannot use all your energy due to facing a dungeon when you only have 1 energy left.
I can pass on feedback about energy carrying over when you restart the Adventure, if you had any left, on the same day.
However, any unused energy does not carry over and is consistent throughout PQ3 game modes.