This question is motivated by this one, where no simple solution within ZFC seems to exist. Let me ask a weaker question then.
Suppose that $K$ is a compact, Hausdorff, non-metrizable space. Does it contain a closed, non-metrizable, totally disconnected subspace?
Note that the cardinality of the (consistent) counterexample in Mathieu Baillif's answer is ludicrously large ($=\omega_{\omega_1}$). Hence my question.
