Dimension bound for normalized orbits in very general abelian varieties

Let $A$ be an abelian variety. A degree $k$ zero-cycle on $A$ is normalized if its sum under the map $\sigma:A^{(k)}\to A$ is zero. Let $Z_A\subset A$ be the locus of points $x$ such that for some effective degree $k$ zero-cycle $Z=x+Z'$, the rational-equivalence orbit $|Z|$ is positive-dimensional and $\sigma(Z)=0$.

Normalized-orbit dimension conjecture. If $A$ is a very general abelian variety, then

$$\dim Z_A\leq k-1.$$

The paper identifies this as the main remaining question behind the conjectural bounds on orbits of zero-cycles. It would imply the preceding gonality conjecture, while the available results provide only partial evidence.