(Non)linear factorization of the Kontsevich tetrahedral flow on the spaces of Poisson structures

In the paper "Formality conjecture" (Ascona'96), Kontsevich introduced a flow on the spaces of
bi-vectors on finite-dimensional affine Poisson manifolds. Quartic in the bi-vectors that evolve, such
flows are linear combinations of two formulas, each determined by a tetrahedron in the graph complex.
Kontsevich claimed that one of the tetrahedral flows preserves the property of Cauchy data to remain
Poisson bi-vectors and that the other flow vanishes identically at every Poisson structure. In the preprint
IHES/M/16/12 (joint with A.Bouisaghouane) it is shown by using 12 counterexamples that both the
claims were false as stated. At the same time, the counterexamples themselves hint that 1:6 is the only
ratio for the Kontsevich tetrahedral flows to preserve the spaces of Poisson bi-vectors. (Other values
were theorized, e.g. 1:(4/3) by Merkulov in arXiv:0809.2385 [math.QA].) The problem to-solve, in
order to prove that the balance 1:6 does the job in the original Kontsevich claim, is the (non)linear
factorization of the Poisson cohomology cocycle condition for the tetrahedral flow via the (poly)differential consequences of the Jacobi identity for the Poisson structure that evolves. This problem's solution
(joint work with R.Buring), phrased in the language of graphs that represent (poly)differential
operators, is the content of the talk.