The original title of this question was "Is there only one (up to homeomorphism) zero-dimensional homogeneous dyadic space of weight $\mathfrak{c}$?". I changed it with the hope of getting a bit more attention.

So I´m interested in the case $\kappa=\mathfrak{c}$ of the following question, asked originally by Efimov in 1965: Is there a zero-dimensional homogeneous dyadic space of weight $\kappa$ that is not homeomorphic to $2^\kappa$?

The only answers I know of to Efimov´s question are:

Pashenkov (1974): Yes, if $\kappa=2^\mu$ for some uncountable $\mu$.

Shapiro (1993) and Bell (1994): No, if $\kappa=\aleph_1$.

So the answer to the question in the original title is Yes when $\mathfrak{c}=\aleph_1$ and it is No when $\mathfrak{c}=2^{\aleph_1}$. Then the real question (perhaps still open) is:

Suppose that $\aleph_1 < \mathfrak{c} > < 2^{\aleph_1}$. Is there a zero-dimensional homogeneous dyadic space of weight $\mathfrak{c}$ that is not homeomorphic to $2^\mathfrak{c}$?

Maybe it is more reasonable to replace $\mathfrak{c}$ by $\aleph_2$ in this question. I would also like to know what happens in general with a $\kappa > \aleph_1$ which is not a power of $2$.

Definitions:
Space = Hausdorff topological space.
$X$ is homogeneous if for all $x,y \in X$ there is an autohomeomorphism of $X$ sending $x$ to $y$.
A space is dyadic if it is the continuous image of some Cantor cube $2^\kappa $.
A space is zero-dimensional if it has a base of clopen subsets.
The weight of a space is the least size of a base for its topology.