Kollár–Szabó's finiteness conjecture for zero-cycles over local fields

Let $X_K$ be a smooth, projective, separably rationally connected variety defined over a local field $K$. Let $d=\dim X_K$, and write $C\!H^d(X_K)_0$ for the degree-zero subgroup, namely the kernel of the degree map $\deg:C\!H_0(X_K)\to\mathbb Z$.

Kollár–Szabó conjecture. The group

$$C\!H^d(X_K)_0$$

is finite.

The conjecture concerns the arithmetic of zero-cycles on rationally connected varieties over local fields. It is known for smooth projective rational surfaces by a theorem of Colliot-Thélène, but remains open in general.