Gersten conjecture for K-theory on Henselian schemes and $\phi$-motivic localisation

A key triviality result for support extension maps for motivic ${\mathbb{A}}^1$-homotopies of cellular motivic spaces $S$ over a DVR spectrum $B$ is proven. Combining with earlier known results on Gersten complex and the K-theory motivic spectrum we achieve a proof of the Gersten Conjecture for K-theory on essentially smooth local Henselian $B$-schemes. Additionally, we outline generalisations for Cousin complexes associated to motivic ${\mathbb{A}}^1$- and $\mathrm{\square }$-homotopies of cellular $B$-spectra. The proof is based on two ingredients: (1) A new ``motivic localisation’’ over $B$, called \emph{$\phi$-motivic}, % localisation giving rise to the $\phi$-motivic homotopy category such that the triviality of the support extension maps and the acyclicity of Cousin complexes hold for all objects $S$, not necessarily cellular. (2) An interpretation of some classes in the motivic ${\mathbb{A}}^1$-homotopies with support defined with respect to the Morel-Voevodsky motivic homotopy category of smooth $B$-schemes in terms of the construction of $\phi$-motivic homotopy category mentioned in Point (1).