It is nearly 85 years since the first ten-square was published. American National Puzzlers' League member Paul M. Bryan (Tunste) exhibited a tautonymic square in the September 1921 Enigma. The example was not perfect, however, with one word (ALALA) being arbitrarily doubled. Furthermore, as tautonymic ten-squares use only five different words, each in two positions, they are not true solutions to the challenge.
"The Best Ten-Squares,"
Word Ways: Vol. 39
, Article 2.
Available at: http://digitalcommons.butler.edu/wordways/vol39/iss2/2