Finite free $p$-Stam extremizer bifurcation conjecture

Let $n\geq 3$ and consider extremizers of the $p$-Stam inequality among normalized monic real-rooted polynomial pairs $(f,g)$ of degree $n$; write $d_{PQ}$ for their pair mismatch and let $\mathrm{He}_n$ denote the degree-$n$ Hermite polynomial. $p$-Stam extremizer bifurcation conjecture. As $p$ decreases from $2$, the extremizers bifurcate: for $p\in(1,2)$, normalized extremizers satisfy $f\not\approx g$, with $d_{PQ}>0$ increasing as $p\to1^+$; each root distribution is bimodal, with two clusters separated by a gap, and this bimodality intensifies as $p\to1^+$; while as $p\to2^-$, the bimodal structure merges, the mismatch vanishes, and the extremizers converge continuously to $(\mathrm{He}_n,\mathrm{He}_n)$. FlowBoost experiments support these structural features, but the precise subcritical extremizer family is explicitly left open and the numerical evidence is not a proof.