Shattering-range conjecture for spherical pure $p$-spin glasses

Let $T_s$ and $T_{sh}$ denote the static and shattering transition temperatures, respectively. For a temperature $T$, say that the free energy landscape is $(E,q,r)$-shattered when it has the shattering property defined in the paper for parameters $(E,q,r)$. Shattering-range conjecture. For every $T$ with

$$T_s\leq T\leq T_{sh},$$

the free energy landscape is $(E,q,r)$-shattered for some $(E,q,r)$, and it is not shattered for any $T>T_{sh}$ or $T<T_s$.

The paper proves shattering on a nontrivial temperature interval beginning no later than $T_s$, while the precise range is expected to be $(T_s,T_{sh}]$. The conjecture asserts both the full shattering interval and its absence outside that interval.