The Hodge-to-de Rham spectral sequence can be arbitrarily non-degenerate in positive characteristic

There is a long tradition of constructing smooth projective varieties in positive characteristic with non-degenerate Hodge-to-de Rham spectral sequences. In this paper, we show that for any $n\geq 1$ and any spot $(i, j)$ on the $E_n$-page, except for the cases where the differential is forced to be zero for trivial reasons, there exists a smooth projective variety whose Hodge-to-de Rham spectral sequence has a non-zero differential with source $E_n^{i, j}$. Our construction is inspired by constructions of nilmanifolds in complex geometry and can be regarded as a positive characteristic analogue.