Linear decay conjecture for strongly Ulam stable groups

Let $G$) be a Strongly Ulam Stable group, and let $F_G$ denote its SUS modulus. Linear decay conjecture. Every Strongly Ulam Stable group $G$ has $O(\varepsilon)$ decay at $0$ of the SUS modulus $F_G$. This would give a uniform linear quantitative bound for the stability of all Strongly Ulam Stable groups; the source presents it as a conjecture motivated by the paper's results and related work, and its resolution is not established here.