Remove the constancy hypothesis on h^2(T) from the family dichotomy theorem

Determine whether the hypothesis that $b\mapsto h^2(Y_b,T_{Y_b})$ is locally constant on closed points can be removed from the family theorem establishing the quasi-$F$-split/Frobenius-split height dichotomy.

Background

The family theorem assumes local constancy of the dimensions h2(Yb,TYb)h^2(Y_b,T_{Y_b}) in order to make the relative lifting obstruction a section of a vector bundle and to construct relative W2W_2-liftings.

The paper explicitly leaves unresolved whether this technical hypothesis is necessary. Removing it would extend the global family dichotomy to broader classes of smooth proper families.

References

We do not know whether Hodge-goodness remains constant there, or whether eq:ct2 can be removed from the family theorem.

eq:ct2:

$b\longmapsto\dim_{\kappa(b)}\HH^2(Y_b,T_{Y_b}) \tag{$\textup{ct}_2$} $

— Quasi-$F$-split primitive symplectic varieties in positive characteristic  (2609.35467 - Zou, 28 Sep 2026) in Remark 5.12, Section 5.4, “The dichotomy on the whole base”