# Which powers of the closed unit interval are homogeneous?

It is known that no finite power of the closed unit interval I is homogeneous, while the countable power, i.e., the Hilbert cube, is. It seems that a power is not homogeneous for every successor cardinal. But, is it homogeneous for every limit cardinal, or is this dependent on some of the more sensitive cardinal properties of the exponent?

For $\kappa \geq \omega$, $I^\kappa$ is homeomorphic to $(I^\omega)^\kappa$, and product of homogeneous spaces is homogeneous, so $I^\kappa$ is homogeneous for every infinite $\kappa$.