Converse composition property for strong CW-regular subdivisions

Determine whether, for strong formal subdivisions X\xrightarrow{\sigma}Y\xrightarrow{\tau}Z between lower Eulerian posets, the assumptions that σ and τ∘σ are strong CW-regular subdivisions imply that τ is a strong CW-regular subdivision when σ has arbitrary rank rather than rank zero.

Background

Lemma~\ref{lem:composeCW} proves the converse implication only under the additional assumption that the first subdivision σ has rank 0. This differs from the corresponding result for strong formal subdivisions, where no rank-zero assumption is required.

The unresolved issue is whether the rank-zero hypothesis can be removed while retaining the implication from σ and τ∘σ being strong CW-regular subdivisions to τ being a strong CW-regular subdivision.

References

Does the converse statement in Lemma~\ref{lem:composeCW} hold without the assumption that $\sigma$ has rank $0$ (c.f. Lemma~\ref{lem:composesfs})? That is, let $\sigma : X \to Y$ and $\tau: Y \to Z$ be strong formal subdivisions between lower Eulerian posets $X$, $Y$, $Z$ with rank functions $\rho_X$, $\rho_Y$, $\rho_Z$ respectively. Assume that $\sigma$ and $\tau \circ \sigma$ are strong $CW$-regular subdivisions. Is $\tau$ necessarily a strong $CW$-regular subdivision?

— Subdivisions of lower Eulerian posets  (2511.16608 - Stapledon, 20 Nov 2025) in Section 10, Section 'Generalizations and further questions'; see also Remark~\ref{rem:converseCW}