Order-$n$ oscillation conjecture for convex domains

Let $K\Subset\mathbb{C}$ be a compact convex domain with nonempty interior, let ${\mathcal P}_n(K)$ denote the polynomials of degree at most $n$ on $K$, and let $\|\cdot\|_{L^q(\partial K)}$ be the $L^q$ norm on its boundary. Order-$n$ oscillation conjecture. For every such $K$ there exists a constant $c_K>0$ such that, for every $p\in{\mathcal P}_n(K)$,

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

The question asks whether nonempty interior is the only additional condition needed for order-$n$ oscillation in the $L^q(\partial K)$ norm. The interval behaves differently, where the order can be as low as $\sqrt{n}$, while the paper establishes order-$n$ lower bounds for some classes of convex domains; the general assertion remains open.