Generalized Bloch conductor formula

Let $A$) be a complete discrete valuation ring with algebraically closed residue field $k$, let $S=\operatorname{Spec}(A)$, and let $p:X\to S$ be a flat, generically smooth, regular $S$-scheme of finite type whose singular locus $Z$ is proper over $S$. Write $s=\operatorname{Spec}(k)$, let $X_s$ be the special fiber, and fix a prime number $\ell$ different from the residue characteristic. Let $\Phi=\Phi_p(\mathbb{Q}_{\ell,X})$ be the $\ell$-adic complex of vanishing cycles. The notation $\Bl(X/S)$ denotes the Bloch intersection number, while $\chi$ and $\operatorname{Sw}$ denote the Euler characteristic and Swan conductor, respectively.

Generalized Bloch conductor formula. If $p:X\to S$ is an arithmetic $S$-scheme with $S$-proper singular locus $Z$, then

$$\Bl(X/S)=-\chi\bigl(\operatorname H^*_{\et}(X_s,\Phi)\bigr)-\operatorname{Sw}\bigl(\operatorname H^*_{\et}(X_s,\Phi)\bigr).$$

This generalizes Bloch's conductor formula from proper arithmetic schemes to arithmetic schemes whose singular locus is proper over the base. It relates an algebro-geometric intersection number to topological and arithmetic invariants of vanishing cycles; the supplied text does not state whether the conjecture is resolved.