NSA for inverse temperatures between saturation and criticality

Let $s$ be a direction and let $\beta_{\rm sat}(s)$ and $\beta_c$ denote the saturation and critical inverse-temperature thresholds, respectively. NSA conjecture. NSA holds for any

$$\beta\in(\beta_{\rm sat}(s),\beta_c).$$

The authors expect this non-saturation property to hold throughout the stated interval, but explicitly note that they do not have a proof.