Esnault–Kerz conjecture on logarithmic ramification bounds

Let $X$ be a normal $\kappa$-scheme of finite type, let $D$ be an effective Cartier divisor on $X$ with smooth complement $U=X-D$, and let $j:U\rightarrow X$ be the canonical injection. Let $\mathscr F$ be a locally constant constructible sheaf of $\Lambda$-modules on $U$, and let $R$ be a Cartier divisor supported on $D$. The ramification of $\mathscr F$ along $D$ is bounded by $R$ in the sense of Deligne if, for every $g:\widetilde C\rightarrow X$ in $\mathcal C(X)$, one has

$$g^*R\geq \mathrm{SW}_{\widetilde C}(g^*(j_!\mathscr F)).$$

Esnault–Kerz conjecture. These conditions are equivalent to requiring that, for every birational morphism $f:X'\rightarrow X$ such that $X'$ is smooth over $\operatorname{Spec}(\kappa)$, $(X'\times_XD)_{\mathrm{red}}$ is a simple normal-crossings divisor, and $f^{-1}(U)=U$, one has

$$f^*R\geq \mathrm{SW}_{X'}(f^*(j_!\mathscr F)).$$

Esnault and Kerz proved that every locally constant constructible sheaf on $U$ has a Deligne-bounded ramification divisor. The conjecture predicts that Abbes–Saito logarithmic ramification detects the sharp lower bound, and is attributed here also to Barr.