On zero-dimensional subspaces of Eberlein compacta

Let us recall that a compact space K is Eberlein compact if it can be embedded into some Banach space X equipped with the weak topology. We will discuss the following problem: Does every nonmetrizable Eberlein compact space contain a closed zero-dimensional, nonmetrizable subspace? We will construct a consistent counterexample. We conjecture that the answer to this problem is independent of ZFC and we will present a partial result justifying this conjecture.