Esnault–Levine–Wittenberg conjecture on ELW-indices over Henselian fields

Let $K$ be the quotient field of an excellent, Henselian discrete valuation ring with algebraically closed residue field $k$, and let $X$ be a proper $K$-scheme. Esnault–Levine–Wittenberg's conjecture. One should have

$$\operatorname{elw}_0(X)=\operatorname{elw}_1(X)=\cdots=\operatorname{elw}_{\dim X}(X).$$

The conjecture was one of the main questions investigated by Esnault, Levine, and Wittenberg and is almost proved in the cited work; the remaining details concern the final step toward equality of all ELW-indices.