Conjecture on bounded doubling of large-spectrum sets

Let $G$ be the ambient finite abelian group, and for a subset $A$ of $G$ let $\text{Spec}_{\varepsilon}(A)$ denote its $\varepsilon$-spectrum. Fix $0<\delta<\alpha<1/2$ and $0<\varepsilon<1/2$. Let $A\subseteq G$ satisfy $|A|\geq |G|^\alpha$.

Bounded-doubling spectrum conjecture. There exists a subset $A'\subseteq A$ with $|A'|\geq |A|/C$ such that

$$|\text{Spec}_{\varepsilon}(A')+\text{Spec}_{\varepsilon}(A')|\leq C|G|^\delta\cdot|\text{Spec}_{\varepsilon}(A')|,$$

where $C=C(\varepsilon,\delta)$.

This conjecture asserts that, after passing to a large subset, the $\varepsilon$-spectrum has bounded doubling up to the factor $|G|^\delta$. The preceding theorem establishes a related result after changing the spectrum threshold from $\varepsilon$ to a smaller value, while the conjecture asks for the conclusion at the original threshold.