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Regularity of T"(X) outside P-spaces and perfectly normal spaces

Determine whether there exists a topological space X that is neither a P-space nor perfectly normal for which the ring T"(X) is Von-Neumann regular.

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Background

The authors show T"(X) is a regular ring when X is a P-space (Theorem 4.1) and when X is perfectly normal (Theorem 4.2). They also provide examples demonstrating that the converses do not hold.

They then ask whether there exist spaces beyond these two classes for which T"(X) is regular.

References

Whether there exists a space X which is neither a P-space nor a perfectly normal space, for which T"(X) is a regular ring remains an undecided problem.

— The ring of real-valued functions which are continuous on a dense cozero set (2502.15358 - Dey et al., 21 Feb 2025) in Paragraph following Remark 4.5, Section 4