Orbit characterization of the filtration on generalized Kummer varieties

Let $X$ be an abelian surface, let $K_n(X)$ be the generalized Kummer variety, and let $z=\{z_0,\ldots,z_n\}\in K_n(X)$ with $z_m\in X$. For a point $z$, let $O_z$ denote its rational-equivalence orbit. Orbit characterization conjecture. One has $\dim O_z\ge n-i$ if and only if there exist points $a_1,\ldots,a_i\in X$ and an integer $k$ such that

$$\sum_{m=0}^n\{z_m\}=\sum_{j=1}^i\bigl(\{a_j\}+\{-a_j\}\bigr)+k\{0\}\in\operatorname{CH}_0(X).$$

In particular, for every $z\in X_0^{n+1}$, there are $a_1,\ldots,a_n\in X$ and $k\in\mathbb{Z}$ satisfying

$$\sum_{m=0}^n\{z_m\}=\sum_{j=1}^n\bigl(\{a_j\}+\{-a_j\}\bigr)+k\{0\}\in\operatorname{CH}_0(X).$$

This gives an explicit geometric reformulation of the proposed filtration description for generalized Kummer varieties. The supplied text does not indicate that this statement has been proved.