The density threshold conjecture for unrestricted sumsets

Let $A\subset\mathbb{N}$, and let $\overline{\mathsf d}(A)$ and $\underline{\mathsf d}(A)$ denote its upper and lower densities.

Density threshold conjecture. The following assertions hold:

1. If $\overline{\mathsf d}(A)>2/3$, then there is an infinite set $B\subset\mathbb{N}$ such that $B+B\subset A$.
2. If $\underline{\mathsf d}(A)>1/2$, then there is an infinite set $B\subset\mathbb{N}$ such that $B+B\subset A$.

Examples in the paper have upper density arbitrarily close to $2/3$ and lower density arbitrarily close to $1/2$ while avoiding such configurations, motivating the proposed sharp thresholds.