Order-$n$ Turán-type oscillation conjecture for convex domains

Let $K\Subset\mathbb C$ be a compact convex domain, let $0<q<\infty$, and let $\mathcal P_n(K)$ denote the polynomials of degree at most $n$ whose zeros lie in $K$. For a polynomial $p$, measure oscillation using the boundary norm $\|p\|_{L^q(\partial K)}$. Order-$n$ Turán-type oscillation conjecture. For every $0<q<\infty$ and every compact convex domain $K\Subset\mathbb C$, there exist $n_0=n_0(q,K)$ and $c_K=C(K,q)>0$ such that, for every $n\ge n_0$ and every $p\in\mathcal P_n(K)$,

$$\|p'\|_{L^q(\partial K)}\ge c_K n\|p\|_{L^q(\partial K)}.$$

The authors note that the conjecture is close to the known $n/\log n$ lower bound, while the interval exhibits smaller-order behavior; the assertion remains open in the stated generality.