Bounded census mass as a function of tip degree

Establish the existence of a function $f:\mathbb{N}\to\mathbb{N}$ such that every tame flow-critical canvas $(G,z)$ satisfies $\sum_{n\in C(G,z)}n\leq f(\deg(z))$.

Background

The paper observes that tame flow-critical canvases may contain arbitrarily many degree-4 vertices, which are therefore omitted from the census. The conjecture asks whether the sum of the remaining, non-4 degrees is bounded solely in terms of the tip degree. The authors later explain that their generation theorem does not provide sufficient control over one alteration operation to prove this conjecture.