Kerz's filtration conjecture for Chow groups of one-cycles

Let $A$ be an excellent henselian discrete valuation ring with uniformising parameter $\pi$ and residue field $k$, let $X$ be a smooth projective scheme over $\operatorname{Spec}(A)$ of relative dimension $d$, and let $X_n=X\times_A A/(\pi^n)$. Let $F_n$ be a filtration of $\mathrm{CH}_1(X)$. Kerz's conjecture. There is a filtration

$$\cdots\subset F_2\subset F_1\subset\mathrm{CH}_1(X)$$

and a well-defined map

$$\mathrm{CH}_1(X)/F_n\leftarrow H^d(X_1,\mathcal{K}^M_{d,X_n})$$

which is inverse to $res_{X_n}$. Furthermore,

$$"\lim_n"F_n\otimes\mathbb{Z}/p^r\mathbb{Z}=0.$$

The conjecture seeks an inverse to the restriction map from Chow groups of one-cycles to the cohomology of infinitesimal thickenings in the mod-$p$ setting. The paper presents it as the mechanism behind injectivity of the restriction map in arbitrary dimension; the relevant injectivity remains open.