Bounded census sum for tame flow-critical canvases

Establish 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

This conjecture asks whether the total of the non-degree-4 entries in the census can be bounded solely as a function of the tip degree. Such a bound would provide structural control over tame flow-critical canvases even when the number of degree-4 vertices is unbounded. The paper later reiterates this conjecture as one of the principal unanswered questions.

References

There exists a function $f: \mathbb{N} \rightarrow \mathbb{N}$ such that if $(G,z)$ is a tame flow-critical canvas, then $$\sum_{n \in C(G,z)} n \leq f(\deg(z)).$$

— Flow-critical graphs  (2502.01451 - Árnadóttir et al., 3 Feb 2025) in Section 1, Subsection 1.2 (Results), immediately after Conjecture \ref{conj:fewlarge}