Equality of T"(X) and T'(X) beyond perfectly normal spaces

Determine whether there exists a topological space X that is not perfectly normal yet satisfies T"(X) = T'(X), where T"(X) is the ring of real-valued functions on X that are continuous on a dense cozero set and T'(X) is the ring of real-valued functions on X that are continuous on an open dense subset.

Background

The paper proves that for perfectly normal spaces X, T"(X) equals T'(X). It also provides an example (the one-point compactification of a discrete real line) in which X is not perfectly normal and T"(X) is a proper subring of T'(X).

This motivates investigating whether equality T"(X) = T'(X) can occur for spaces outside the perfectly normal class.