Esnault–Kerz conjecture on curvewise computation of the Swan divisor

Let $X$ be a normal $\kappa$-scheme of finite type, let $D$ be an integral effective Cartier divisor 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 $\mathcal C(X)$ be the set of canonical morphisms $g:\widetilde C\rightarrow X$ from normalizations of suitable one-dimensional integral closed subschemes. Define $\mathcal{PC}(X)$ as the set of pairs $(g:\widetilde C\rightarrow X,x)$ with $g\in\mathcal C(X)$ and $x$ a closed point of $g^*D$. Write $m_x(g^*D)$ for the multiplicity of $g^*D$ at $x$, and let $\mathrm{sw}_D(j_!\mathscr F)$ be the coefficient of the Swan divisor after replacing $X$ by a smooth open neighborhood of the generic point of $D$.

Esnault–Kerz conjecture. One has

$$\sup_{\mathcal{PC}(X)}\frac{\mathrm{sw}_x(g^*j_!\mathscr F)}{m_x(g^*D)}=\mathrm{sw}_D(j_!\mathscr F).$$

The conjecture asserts that the generic coefficient of the Swan divisor is recovered exactly as the supremum of normalized curvewise Swan conductors. It is the sharp curve-restriction formulation of the logarithmic ramification prediction described in the source.