Large-prime strong-counterexample conjecture for abelian surfaces

Let $K$ be a number field, let $\ell$ be a prime, and let $(A,\ell)$ be a strong counterexample to the local-global principle for isogenies of prime degree between abelian surfaces, meaning that $A$ has a locally everywhere isogeny of degree $\ell$ but no global one. Large-prime strong-counterexample conjecture. There is a constant $b=b(K)$ such that, for all primes $\ell>b(K)$ and all such strong counterexamples $(A,\ell)$, the abelian surface $A$ is geometrically isogenous to the square of an elliptic curve with complex multiplication. This predicts that sufficiently large-prime strong counterexamples can occur only in the CM-square case, after the paper excludes the other relevant monodromy cases for all but finitely many primes.