Equivalence of formal K-flatness and K-flatness

Let $F$ be a generically flat family of pure, coherent sheaves of relative dimension $d$ on ${\mathbb P}^n_S$. Formal K-flatness means that the relevant K-flatness condition holds after completion at every point, while K-flatness is the corresponding global condition.

Formal K-flatness conjecture. Formal K-flatness is equivalent to K-flatness.

The paper has established implications from formal K-flatness to K-flatness and from K-flatness to stable C-flatness, and proves that K-flatness is equivalent to stable C-flatness. The converse implication from K-flatness to formal K-flatness is presented as likely to follow from the methods, but is not established here.