The well-definedness conjecture for the pre-foam evaluation

Let $S$ be a closed pre-foam, namely an abstract finite $2$-dimensional cell complex with foam local structure, possibly with tetrahedral points and dots. Define $\Jflat(S)\in\F$ by the stated rules: discard non-bipartite seam graphs, cancel paired tetrahedral points, impose the monodromy condition, use neck-cutting to reduce to theta-foams and closed surfaces, and evaluate the resulting components by the specified rules. The well-definedness conjecture. The above rules lead to a unique, well-defined evaluation $\Jflat(S)\in\F$ for every pre-foam $S$.