Optimal covering conjecture for small-doubling sumsets

Let $A$ be a set of size $s$ with doubling constant bounded by $K$, meaning $|A+A|\leq K|A|$. For each $ell$ let $\mathcal{F}_\ell$ be a collection of sets, and let $\tilde{O}$ and $\tilde{\Omega}$ suppress polylogarithmic factors in the relevant parameter. Optimal covering conjecture. There exist collections of sets $\mathcal{F}_\ell$ such that

$$\log |\mathcal{F}_\ell|=\tilde{O}(2^\ell),$$

and

$$\min_{F\in\mathcal{F}_\ell}|F|=\tilde{\Omega}(2^\ell s),$$

such that for every $A$ with $|A|=s$ and $|A+A|\leq K|A|$, there exists $\ell\leq\log_2 K$ and $F\in\mathcal{F}_\ell$ satisfying $A+A\supseteq F$. An optimal dependence on the doubling parameter would provide sharp obstructions for the independence number of sparse random Cayley graphs, up to logarithmic factors. The source gives no resolution of this conjecture.