Let $A$ be an abelian variety over an algebraically closed field $k$. 

Let $\phi:A\to A$ be an étale isogeny (over $k$). Suppose that the set $\cup_{r\geq 0}({\rm ker}\,\phi^{\circ r})(k)$ is 
Zariski dense in $A$. Here $\phi^{\circ r}:=\phi\circ\dots\circ\phi$ ($r$-times).

Now let $\lambda:A\to A$ be an endomorphism and suppose that for all $r\geq 0$, we 
have $$\lambda(({\rm ker}\,\phi^{\circ r})(k))\subseteq ({\rm ker}\,\phi^{\circ r})(k).$$

Does it follow that $\lambda\circ \phi=\phi\circ\lambda$? (in other words: 
does it follow that $\lambda$ and $\phi$ commute?).

One may of course ask the same question for any isogeny (not just étale) and 
formulate a more general condition involving finite group schemes but the case above is the case I am interested in.