Let $A/k$ be an abelian variety over a number field $k$ with a polarization of minimal degree $d>1$. (Assume all Tate-Shafarevich groups to be finite.)

What can one say about the order of $\mathrm{III}(A/k)$ in terms of being a multiple of a square?

Is it true that the order of $\mathrm{III}(A/k)$ equals $km^2$ for some $k$ dividing $2d$? If yes, why? Is this even true if $A/k$ is not isogenous to a principally polarized abelian variety?