I can't find anything on the NCAA website that confirms it, but this article by Mechelle Voepel from November 2021 says starting in 2023 the opening rounds will be at a neutral site to be determined. Maybe they will go back to 8 neutral sites similarly to how they did in 2005-2008 (but schools could host on their home floor if they won the bid in 2005-2008)?
abcnews.go.com
Those 4 years the places UConn played were:
2005- Storrs, Regionals in Kansas City, FF in Indianapolis. Lost to Stanford in Sweet 16.
2006- University Park, PA, Regionals in Bridgeport, FF in Boston. Lost to Duke in Elite 8.
2007- Hartford, Regionals in
Fresno CA, FF in Cleveland. Lost to LSU in Elite 8.
2008- Bridgeport, Regionals in Greensboro, FF in Tampa. Lost to Stanford in Final Four.
Currently teams can't play on a court they played more than 3 games in the regular season during the NCAA tournament. So maybe Uconn will be playing the first two rounds in Bridgeport or Albany starting next year. Are there enough hotel rooms near the Mohegan Sun Arena to host 8 teams?