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Expressing T"(X) as C(X)^{P} for an ideal of closed sets

Investigate whether for every topological space X there exists an ideal P of closed subsets of X such that T"(X) equals C(X)^{P}, the ring of real-valued functions on X that are continuous outside members of P.

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Background

The authors note that for perfectly normal spaces X, T"(X)=T'(X)=C(X){P} with P the collection of closed nowhere dense sets, and for almost P-spaces X, T"(X)=C(X)=C(X){P} with P={∅}.

They speculate whether such a representation holds in general for T"(X).

References

See that for a perfectly normal space X, T"(X) = T'(X) = C(X)p, where P is the collection of all closed nowhere dense sets and for an almost P-space X, T"(X) = C(X) = C(X)p, where P = {0}. We speculate whether T"(X) can always be expressed as C(X)p for some ideal P of closed subsets of X.

— The ring of real-valued functions which are continuous on a dense cozero set (2502.15358 - Dey et al., 21 Feb 2025) in Section 6, item (8)