I believe the full picture to be rather open, but Takayuki Kihara has shown that there is a computable embedding of $K^4$ into $K^3$ in

[T. Kihara: *The Brouwer invariance theorems in reverse mathematics* arXiv 2002.10715][1]

The argument is outlined in the middle of p. 13. It uses Ovrekov's "constructive map of the square into itself, which moves every constructive point", from which it follows that $\mathrm{WKL}$ fails for the computable numbers. Kihara's paper then shows that for any model of $\mathbb{R}$ in which $\mathrm{WKL}$ fails, there is a topological embedding from $\mathbb{R}^m$ into $\mathbb{R}^3$ for any $m$.

Whether we can even have a computable injection of $K^3$ into $K^2$ is listed as an open question here:

http://www.math.mi.i.nagoya-u.ac.jp/~kihara/questions.html

Although I don't have much to go on with, I suspect that asking about mutual embeddability might be the cleaner question here compared to computable homeomorphism.


  [1]: https://arxiv.org/abs/2002.10715