Existence of a homogeneous pseudocompact space with non-pseudocompact square

Establish whether there exists a homogeneous pseudocompact Tychonoff space X such that the product X × X is not pseudocompact.

Background

The paper reviews product behavior of pseudocompact spaces. While products of pseudocompact topological groups and Mal’cev spaces remain pseudocompact, there exist homogeneous pseudocompact spaces X and Y with X × Y not pseudocompact. The special case of squares of homogeneous pseudocompact spaces remains unsettled.

The authors explicitly point out that the question of whether a homogeneous pseudocompact space can have a non-pseudocompact square is still open, referencing Question 5.3 of Comfort–van Mill (1985).

References

However, the problem of the existence of a homogeneous pseudocompact space $X$ for which the product $X \times X$ is not pseudocompact Question 5.3 still remains open.

— Extensions and factorizations of topological and semitopological universal algebras (2402.01418 - Reznichenko, 2 Feb 2024) in Section 3.1 (Main results: Extension of operations on X), paragraph discussing products of pseudocompact spaces