Characterization of Eberlein compacts via embeddings into Cp(X,Y) with non-metrizable Y

Determine whether there exists a non-metrizable topological space Y such that a compact space K is an Eberlein compact if and only if K homeomorphically embeds into Cp(X,Y), the space of continuous functions from a compact X to Y with the pointwise topology, for some compact X.

Background

The paper shows that if Y is metrizable (in fact containing a copy of R), then any compact subspace of Cp(X,Y) is an Eberlein compact (Proposition 4.1), providing one direction of a characterization.

This problem asks whether such a characterization of Eberlein compacta via embeddings into Cp(X,Y) can be achieved with a non-metrizable codomain Y, i.e., whether there exists a non-metrizable Y for which the equivalence between being Eberlein and embeddability into some Cp(X,Y) holds.