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.