The flip-width subpolynomial collapse conjecture

Let $\mathcal C$ be a hereditary graph class. For each $r\in\mathbb N$, let $\operatorname{fw}_r(G)$ denote the flip-width of $G$ for robber speed $r$.

Flip-width subpolynomial collapse conjecture. The following conditions are equivalent:

1. $\mathcal C$ has almost bounded flip-width, that is, for every fixed $r\in\mathbb N$ and $\varepsilon>0$,

$$\operatorname{fw}_r(G)=o(|G|^\varepsilon)$$

for all $G\in\mathcal C$; 2. for every fixed $r\in\mathbb N$,

$$\operatorname{fw}_r(G)=o(|G|^{1/2})$$

for all $G\in\mathcal C$.

This predicts a collapse from the subpolynomial bound at exponent $1/2$ to subpolynomial bounds at every positive exponent. The source presents it as a conjecture supported by the paper's evidence; its general validity remains open.