The 1-invariance conjecture for tame cohomology

Let $X$ be a scheme over a base scheme $S$, and let $pr:\mathbb A^1_X\to X$ be the natural projection. For a torsion sheaf $F\in \operatorname{Sh}_t(X/S)$, write $H^q_t(X/S,F)$ for tame cohomology. The \mathbb A^1-invariance conjecture. For every such $F$, the natural map

$$H^q_t(X/S,F)\to H^q_t(\mathbb A^1_X/S,pr^*F)$$

is an isomorphism for all $q\geq 0$.

This predicts homotopy invariance for tame cohomology. It is known in the stated paper under stronger hypotheses, including affine noetherian positive-characteristic bases, regular schemes essentially of finite type, resolution of singularities, and locally constant torsion coefficients; the general assertion remains open.