Stein's conjecture on non-square parts of Tate–Shafarevich groups

Let $A/\mathbb{Q}$ range over all abelian varieties over $\mathbb{Q}$, and suppose that $\sha(A/\mathbb{Q})$ is finite so that its order has a well-defined non-square part. Stein's conjecture. As one ranges over all abelian varieties $A/\mathbb{Q}$, every square-free natural number appears as the non-square part of the order of some $\sha(A/\mathbb{Q})$. This conjecture asks how broadly non-square orders of finite Tate–Shafarevich groups can occur; the preceding results give restrictions on such non-square parts and provide examples for many odd primes, but the stated assertion is not resolved here.