Sharpness of the logarithmic inverse-temperature threshold for uniform separation

Let $(\beta_n)$ be a sequence of inverse temperatures, and consider the Coulomb gas configurations at these temperatures. Sharpness conjecture. If almost sure uniform separation holds for some sequence of inverse temperatures $(\beta_n)$, then necessarily

$$\beta_n\ge c\log n$$

for some constant $c>0$. This predicts that the logarithmic scale in the uniform-separation theorem is necessary, although the statement does not specify the dependence of $c$ on the model or the precise meaning of uniform separation beyond the referenced theorem.