It is known that no two distinct finite powers of the closed unit interval are homeomorphic:

$I^m$ is homeomorphic to $I^n$ iff $m=n$. (Brouwer, Lebesgue, 1911)

Is the analogous result for infinite powers of $I$ true?

That is, is it true that $I^\alpha$ is homeomorphic to $I^\beta$ iff $\alpha=\beta$?

(An affirmative answer would give a hope of defining a transfinite inductive dimension for all spaces, a problem essentially posed by Carl Menger.)

If not, what are the homeomorphism classes among the infinite powers of $I$?