The conjecture that flat finite-type morphisms of Noetherian schemes are very flat

Let $f:Y\longrightarrow X$ be a flat morphism of finite type between Noetherian schemes. A morphism is very flat when, locally on affine open subschemes, the corresponding ring map makes the target ring a very flat module over the source ring.

Very-flatness conjecture. Every flat morphism of finite type between Noetherian schemes is very flat.

This conjecture is presented as natural, with evidence discussed later in the paper. Its status is not determined by the supplied text.