Which compacta contain copies of Cantor cubes?

Question

It is well-known that each uncountable compact metrizable space $X$ contains a homeomorphic copy of the Cantor cube $\{0,1\}^\omega$. What about copies of Cantor cubes of larger weight?

Problem. Does every uncountable Dugundji compact space $X$ contain a topological copy of the Cantor cube $\{0,1\}^{\kappa}$ of weight $\kappa=w(X)$ ?

A compact Hausdorff space $X$ is Dugundji if and only if $X$ is an AE(0)-space, which means that each continuous map $f:B\to X$ defined on a closed subspace $B$ of a zero-dimensional compact Hausdorff space $A$ admits a continuous extension $\bar f:A\to X$.

Answer 1

It was proved independently by Efimov and Gerlits that the answer is yes if $\kappa$ has uncountable cofinality. In fact they proved this for any dyadic $X$ (it is well known that Dugundji compacta are dyadic). Their theorem (see "Mappings and imbeddings of dyadic spaces" by Efimov or "On subspaces of dyadic compacta" by Gerlits) actually says:

For a dyadic $X$ with $w(X)=\kappa$ the following are equivalent:

1) $X$ contains a copy of $2^\kappa$.

2) $X$ maps continuously onto $[0,1]^\kappa$.

3) $X$ is not a countable union of closed subspaces of weight less than $\kappa$.

I don't know if this can be improved at all using the stronger hypothesis of $X$ being Dugundgi.