In 2006-176, Ross Eckler stated that a heterogrammatic 5-square is impossible to achieve, referring I believe to double squares, but without giving a reason. In the May 1979 Colloquy (following an earlier effort February 1978), Jeff Grant had presented a double 5-square with 22 different letters, all ten words being in the Oxford English Dictionary.
"Near-Heterogrammatic Double 5-Squares,"
Word Ways: Vol. 40
, Article 4.
Available at: https://digitalcommons.butler.edu/wordways/vol40/iss3/4