<Solid-2024-248>, hint

Suppose TB=TC, isosceles, and <CTD=90, CTD is right triangle

and TT' is height of TABCD, O is center of ABCD, M,N  the midpoint of BC,AD.

O' is the center of circle CTD.

Then T' be on the segment MN, BM=MC=2=AN=ND

Since CTD is right triangle O'C=O'D=O'T=2. In triangle TT'O' , <TT'O'=90

TO=2, T'O'>2, OO'=2 which is contradiction. Hence ATD is right isoscele triangle, TA=TD=TN=2

and so TABCD has max volume when TT'=2=TN (4x4x2)/3=32/3