Constancy of Hodge-goodness at the boundary case n=p

Determine whether Hodge-goodness remains constant in smooth proper families of relative dimension $2n$ when $n=p$, under the family hypotheses used in the paper.

Background

The paper proves constancy of Hodge-goodness under relative $W_2$-liftings when $n<p$. The proof uses low-degree Deligne–Illusie decompositions and a cup product from degrees below $p$.

At the boundary case $n=p$, the coherent cohomology dimensions remain constant, but the cup product from degree $p-1$ to degree $p+1$ lies outside the range controlled by the argument. The authors therefore leave unresolved whether Hodge-goodness is nevertheless constant.