Brosnan's p-incompressibility conjecture for multiplication maps on abelian varieties

Let $A$ be an abelian variety over a field $L$ of characteristic zero, and let $[p]:A\to A$ denote multiplication by a prime $p$. Brosnan's conjecture. For every prime $p$, the map $[p]$ is $p$-incompressible. This conjecture predicts a uniform incompressibility property for multiplication maps on abelian varieties; the source presents subsequent results as progress toward it, but does not establish the full statement.