Finite free $p$-Stam phase transition conjecture

For each $n\geq 3$, consider the $p$-Stam inequality for monic real-rooted pairs of degree $n$, with critical exponent denoted by $p^*$. $p$-Stam phase transition conjecture. For all $n\geq 3$, the $p$-Stam inequality holds for all monic real-rooted pairs if and only if

$$p\in(1,2],$$

so that $p^*=2$, independently of $n$. The paper proves failure at the Hermite pair for every $p>2$ and presents computational and analytic evidence for the threshold; validity throughout $(1,2]$ remains open.