Rankin's set conjecture for the maximal density of geometric-progression-free sets

Let $G^*_3$ denote Rankin's geometric-progression-free set, and let $d(G^*_3)$ be its density. Let $\alpha$ be the supremum of the densities of geometric-progression-free sets that possess a density.

Rankin's set conjecture. Rankin's set $G^*_3$ has the largest possible density among geometric-progression-free sets which have a density, so

$$\alpha=d(G^*_3).$$

The paper establishes an effective method for computing the maximal upper density of sets avoiding three-term geometric progressions, but the precise value of $\alpha$ remains open. This conjecture is stronger than the question of whether $\alpha$ is strictly smaller than the corresponding maximal upper density.