The power-of-p conjecture for Hochschild-Kostant-Rosenberg differentials

Let $A$ be an $\mathbb F_p$-algebra and let $X/A$ be a scheme. Consider the Hochschild-Kostant-Rosenberg spectral sequence for $X/A$, with differentials denoted by $d_r$. Power-of-$p$ conjecture. The differentials $d_r$ are zero unless $r$ is a power of $p$. The theorem immediately preceding this conjecture proves vanishing unless $r\equiv 1\pmod{p-1}$, while the example $X=B\mu_{p^n}$ shows nontrivial differentials on page $p^n$; whether all remaining possible nonzero differentials occur only on pages that are powers of $p$ is left open.