p-independence of parity for abelian-variety Selmer ranks

Let $K$ be a number field, let $A/K$ be an abelian variety, and for each prime $p$ let $\selrk{p}(A/K)$ denote the $p$-Selmer rank. p-independence conjecture. The parity of $\selrk{p}(A/K)$ is independent of $p$.

This is a proposed general parity principle for abelian varieties. The source states it as an assumption that could allow the paper's methods to relax the requirement that $F/K$ be a $p$-extension; no resolution is given.