Does Cp(X,2) exponentially separable characterize Corson compacta?

Determine whether every zero-dimensional compact Hausdorff space X for which the function space Cp(X,2)—the set of all {0,1}-valued continuous functions on X endowed with the topology of pointwise convergence—is exponentially separable must be a Corson compactum.

Background

The paper surveys when function spaces Cp(X,2) exhibit exponential separability, providing several equivalences for zero-dimensional compact spaces, including that Cp(X,2) is exponentially separable for Corson compacta and that Cp(X,2) is a countable union of compact scattered spaces precisely for Eberlein compacta. A summary diagram indicates that most implications are equivalences except for one direction.

This leads to the natural problem of identifying whether the exponential separability of Cp(X,2) characterizes Corson compacta, i.e., whether the converse to known results holds.