Voisin's strong orbit conjecture for very general abelian varieties

Let $k\in\mathbb{Z}_{>0}$, let $A$ be a very general abelian variety, and let $\mathscr{G}(k)$ be the least dimension such that a very general abelian variety of that dimension has no positive-dimensional orbit of degree $k$. Voisin's strong orbit conjecture. A very general abelian variety of dimension at least $k+1$ does not have a positive-dimensional orbit of degree $k$, equivalently,

$$\mathscr{G}(k)\leq k+1.$$

The source says that Voisin's dimension conjecture for normalized positive-dimensional orbits implies this stronger statement, which would substantially improve the previously discussed bounds.