Census bound for tame flow-critical canvases

Prove that every tame flow-critical canvas $(G,z)$ satisfies $\sum_{n\in C(G,z)}(n-6)\leq \deg(z)-8$.

Background

A census $C(G,z)$ records the degrees, other than 4, of vertices distinct from the tip $z$ in a canvas. The paper proves strong restrictions on the largest census entries, including that a vertex of degree $\deg(z)-2$ forces a highly constrained census. This conjecture seeks a global bound on the total excess of census degrees above 6 and would imply a weaker density bound for flow-critical graphs.