Sharp width-dependent Turán conjecture for convex domains

Let $K\Subset\mathbb C$ be a compact convex domain. Denote by $d$ its diameter and by $w$ its width, let $n\in\mathbb N$, and let $\mathcal P_n(K)$ denote the relevant class of polynomials of degree at most $n$ on $K$. For $q\ge 1$, write $\|\cdot\|_{L^q(\partial K)}$ for the boundary $L^q$ norm.

Width-dependent Turán conjecture. There exists an absolute constant $c>0$ such that, for every compact convex domain $K\Subset\mathbb C$ 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 would sharpen the conjectured linear lower bound by identifying the geometric dependence through the width-to-diameter-squared ratio. The supplied passage presents it as a further sharpening motivated by the maximum-norm result and constructions giving matching order, but it does not establish the claim.