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Converse of Proposition 5.4 (countable pseudocharacter vs. nowhere almost P-space)

Ascertain whether every Tychonoff nowhere almost P-space has countable pseudocharacter; equivalently, determine whether there exists a Tychonoff nowhere almost P-space that is not of countable pseudocharacter.

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Background

Proposition 5.4 proves that any Tychonoff space with countable pseudocharacter is a nowhere almost P-space.

The authors state they were unable to prove or disprove the converse and refer the question to the concluding section.

References

We have certainly pondered whether the converse of Proposition 5.4 holds true. We have however neither been able to prove nor disprove the converse. This question is stated at the end of this article and is left for the readers.

— The ring of real-valued functions which are continuous on a dense cozero set (2502.15358 - Dey et al., 21 Feb 2025) in Remark 5.7, Section 5