Small-doubling structural conjecture in cyclic groups

Let $n$ be a positive integer, let $\mathbb{Z}_n$ be the cyclic group of order $n$, and let $A\subseteq\mathbb{Z}_n$. For a subset $S\subseteq\mathbb{Z}_n$, write $2S=S+S$. Small-doubling structural conjecture. For every $\varepsilon>0$, there exist positive constants $C_1(\varepsilon)$ and $C_2(\varepsilon)$ such that, if

$$|A|<C_1(\varepsilon)^{-1}n\qquad\text{and}\qquad |2A|<(3-\varepsilon)|A|,$$

then there exist a subset $P\subseteq\mathbb{Z}_n$ and a proper subgroup $H<\mathbb{Z}_n$ such that

$$\frac{|2P|}{|P|}\leq\frac{|2A|}{|A|},$$ $$A\subseteq P+H,$$ $$(|2P|-|P|)|H|\leq |2A|-|A|,$$

and either $|P|\leq C_2(\varepsilon)$ or $P$ is an arithmetic progression. The conjecture seeks a structural description of sufficiently sparse subsets of cyclic groups with doubling constant below $3-\varepsilon$, improving on the sharp threshold $9/4$ established in the surrounding theorem.