Asymptotic sharpness of the Bernstein inequality constant for rational functions in Hardy spaces

Let $n\geq 1$, $r\in[0,1)$, and let $\mathcal{R}_{n,r}$ be the rational functions $p/q$ in the unit disc, where $p,q$ are analytic polynomials of degree at most $n$, $\operatorname{deg}p<\operatorname{deg}q$, and every zero of $q$ lies outside $(1/r)\mathbb{D}$. For Banach spaces $X,Y$ of holomorphic functions on the unit disc, let $\mathcal{C}_{n,r}(X,Y)$ be the best constant such that

$$\|f'\|_X\leq \mathcal{C}_{n,r}(X,Y)\|f\|_Y\qquad(f\in\mathcal{R}_{n,r}).$$

Asymptotic sharpness conjecture. For $1\leq p\leq\infty$, the constant $(1+r)^{1/p}$ in the estimate

$$\mathcal{C}_{n,r}(H^1,H^p)\leq (1+r)^{1/p}\frac{n}{(1-r)^{1/p}}$$

should be asymptotically sharp as $n\to+\infty$. In particular, the paper records the known result

$$\lim_{n\to\infty}\frac{\mathcal{C}_{n,r}(H^2,H^2)}{n}=\frac{1+r}{1-r}.$$

The claim concerns the optimal asymptotic constant in Bernstein-type inequalities for rational functions with poles constrained outside a larger disc. The cited $H^2$ limit is stated as already proved, while the general asymptotic sharpness is presented as something that should be proved using the same test function.