The Gersten conjecture for tame logarithmic de Rham–Witt sheaves

Let $k$ be a perfect field of characteristic $p>0$, and let $X$ be a smooth $d$-dimensional $k$-scheme. Write $X^{(a)}$ for the points of codimension $a$, $i_{*,x}$ for extension by zero from the point $x$, $\beta$ for the morphism from the tame site to the relevant underlying site, and $W_r\Omega_{X,\log,t}^n$ for the tame logarithmic de Rham–Witt sheaf. Gersten conjecture. There is an exact sequence of sheaves

$$0\to R^j\beta_*(W_r\Omega_{X,\log,t}^n)\to\bigoplus_{x\in X^{(0)}}i_{*,x}H^j(x,W_r\Omega_{x,\log,t}^n)\to\bigoplus_{x\in X^{(1)}}i_{*,x}H^j(x,W_r\Omega_{x,\log,t}^{n-1})\to$$ $$\dots\to\bigoplus_{x\in X^{(d)}}i_{*,x}H^j(x,W_r\Omega_{x,\log,t}^{n-d})\to0.$$

This is proposed as the optimal Gersten-type resolution in the tame topology. The source relates it to a purity conjecture and gives no resolution.