Uniform geometric Turán-type oscillation conjecture for convex domains

Let $K\Subset\mathbb C$ be a compact convex domain with width $w$ and diameter $d$, and let $\mathcal P_n(K)$ denote the polynomials of degree at most $n$ whose zeros lie in $K$. For $0<q<\infty$, measure the polynomial and its derivative on the boundary using the norm $\|\cdot\|_{L^q(\partial K)}$. Uniform geometric Turán-type oscillation conjecture. There exists an absolute constant $c>0$ and an $n_0=n_0(q,w,d)$ such that, for every $0<q<\infty$, every compact convex domain $K\Subset\mathbb C$, every $n\ge n_0$, and every $p\in\mathcal P_n(K)$,

$$\|p'\|_{L^q(\partial K)}\ge c\frac{w}{d^2}n\|p\|_{L^q(\partial K)}.$$

This is a more precise uniform version of the order-$n$ conjecture, retaining explicit dependence on the width and diameter; the paper presents it as open and notes that only an $n/\log n$ lower bound is currently available in full generality.