Existence of Polish spaces realizing Rosenthal compacts inside Scro(X×Y) and Scru(X×Y)

Determine whether, for any Rosenthal compact K, there exist Polish spaces X and Y such that K homeomorphically embeds into the space S(X×Y) of separately continuous real-valued functions with the cross-open topology Scro(X×Y), and whether the same holds for the cross-uniform topology Scru(X×Y).

Background

Rosenthal compacts are classical compacta arising as compact subsets of spaces of Baire-one functions over Polish domains. The authors show that only metrizable compacta embed into S([0,1]^2) and provide a general characterization for compact X and Y (Theorem 7.1).

This problem asks whether Rosenthal compacta can be realized as compact subspaces within the spaces of separately continuous real-valued functions over Polish (non-compact, per the authors’ remark) domains equipped with the cross-open or cross-uniform topologies.