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Existence of X with T"(X) not almost regular

Determine whether there exists a topological space X for which the ring T"(X) is not almost regular, i.e., contains an element that is neither a unit nor a zero divisor.

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Background

The authors show that T"(X) is almost regular when X is perfectly normal (since T"(X)=T'(X) is regular), when X is a nowhere almost P-space, and when X is an almost P-space (since T"(X)=C(X) is almost regular).

They ask whether there exists any space X for which T"(X) fails to be almost regular.

References

When X is perfectly normal, T"(X) = T'(X) which is a regular ring, and hence an almost regular ring as well. Furthermore, when X is a nowhere almost P- space, we have seen that T"(X) is almost regular. Moreover, when X is an almost P-space, T"(X) = C(X) which is an almost regular ring. So we raise the question: Does there exist a topological space X for which T"(X) is not an almost regular ring?

— 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 (9)