The odd Vassiliev conjecture at four loops

I will outline a proof that odd Jacobi diagrams vanish up to four loops, extending a result of Moskovich and Ohtuski by one. My idea is to simplify calculations of Jacobi diagrams with a limited number of loops with the formalism of graph homology with polynomial coefficients, and using operadic composition of Jacobi diagrams with leaves. The calculation for three loops becomes much shorter, while the calculation for four loops becomes feasible without computer assistance.