ZFC existence of a Corson compactum whose countable power lacks a dense metrizable subspace

Determine whether, within ZFC, there exists a Corson compact space X such that its countable power X^ω does not contain a dense metrizable subspace.

Background

The paper establishes a characterization for when the countable power of a Corson compactum has a dense metrizable subspace, namely that X^ω has such a subspace if and only if c(X^ω) > w(X). Using this, the authors construct several consistent examples of Corson compacta whose countable powers do not have dense metrizable subspaces.

Despite these consistent constructions, the authors note that it remains unresolved in ZFC whether any Corson compactum exists whose countable power lacks a dense metrizable subspace. This question sits at the core of the paper’s theme of understanding dense metrizable subspaces in powers of Corson compacta.