Smoothness of flag varieties for non-tame posets

Determine whether there exists a finite non-tame poset P such that every poset flag variety Fl_P(\mathbf n;n) is smooth for all dimension vectors \mathbf n and ambient dimensions n.

Background

The paper proves that tame finite posets have smooth poset flag varieties, realized as iterated Grassmannian bundles, and admits affine pavings. It also gives examples of non-tame posets whose flag varieties can be singular. The authors leave unresolved whether non-tameness necessarily prevents smoothness in at least one dimension vector.

References

It remains an open problem whether there exists a non-tame poset $P$ such that $Fl_P(\mathbf{n};n)$ is smooth for all $(\mathbf{n};n)$.

— Quot scheme of points on torus knot singularities  (2608.16086 - Huang et al., 17 Aug 2026) in Section 6, subsection “Poset flag varieties on tame posets”