The relative Stiefel–Whitney and Hasse–Witt class conjecture

Let $S$ be a normal divisorial scheme over $\mathbb Z[\frac1{2\ell}]$, either separated or noetherian, and let $f:X\to S$ be a proper smooth morphism of relative even dimension $n$. Define

$$\eta=\sum_{q<\frac n2}(-1)^q\left(\frac n2-q\right)\operatorname{rank} Rf_*\Omega^q_{X/S}.$$

Let $c_2,c_\ell\in H^2(S,\mathbb Z/2\mathbb Z)$ be the characteristic classes defined in the source. The relative Stiefel–Whitney–Hasse–Witt conjecture. One has

$$sw_2(H^\bullet_\ell(X/S))=hw_2(H^\bullet_{dR}(X/S))+\{2,hw_1(H^\bullet_{dR}(X/S))\}+\eta\cdot(c_\ell-c_2)$$

in $H^2(S,\mathbb Z/2\mathbb Z)$. This conjecture proposes a comparison between the second Stiefel–Whitney class of relative $\ell$-adic cohomology and the second Hasse–Witt class of relative de Rham cohomology, with correction terms determined by the first Hasse–Witt class and the characteristic classes $c_\ell,c_2$.