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

The setup is to show to analyze how exactly the Brouwer Invariance Theorem fails in models of $\mathrm{RCA}_0 + \neg\mathrm{WKL}$, and apparently the 4 to 3 embedding is what pops out. Since the computable reals are a model of that theory, we get the consequence for them. 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.

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

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.