  We adapt Wei Zhang's proof of Kolyvagin's conjecture for modular abelian
varieties over $\mathbb{Q}$ to rely on the BDP main conjecture instead of on
the cyclotomic main conjecture. The main ingredient is a reduction to a case
that is tractable by the BDP main conjecture, in a similar spirit to Zhang's
reduction to the rank one case. By using the BDP main conjecture instead of the
cyclotomic main conjecture, our approach is more suitable than Zhang's to
extend to modular abelian varieties over totally real fields.
