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 $\bar f:A\to X$.