Non-metrizable compact subspaces within Scro(X×Y) and Scru(X×Y) over Polish spaces

Ascertain whether there exist Polish spaces X and Y such that the space S(X×Y) of separately continuous real-valued functions equipped with the cross-open topology Scro(X×Y) (respectively the cross-uniform topology Scru(X×Y)) contains a non-metrizable compact subspace K.

Background

The authors prove that for infinite metrizable compact X and Y, compact subspaces of S(X×Y,Z) are necessarily metrizable (Corollary 7.2).

This problem investigates whether allowing Polish (necessarily non-compact, per the authors’ remark) domains X and Y permits the presence of non-metrizable compact subspaces within the spaces of separately continuous functions with the cross-open or cross-uniform topologies.