Interrelation between Nikolskii–Bernstein constants for trigonometric polynomials and entire functions of exponential type

Let $0<p\le \infty$, $\mathcal{C}(n;p;r)=\sup_{T}\frac{\|T^{(r)}\|_{L^{\infty}[0,2\pi)}}{\|T\|_{L^{p}[0,2\pi)}}$ and $\mathcal{L}(p;r)=\sup_{F}\frac{\|F^{(r)}\|_{L^{\infty}(\mathbb{R})}}{\|F\|_{L^{p}(\mathbb{R})}}$ be the sharp Nikolskii–Bernstein constants for $r$-th derivatives of trigonometric polynomials of degree $n$ and entire functions of exponential type $1$ respectively. Recently E. Levin and D. Lubinsky have proved that for the Nikolskii constants
$$ \mathcal{C}(n;p;0)=n^{1/p}\mathcal{L}(p;0)(1+o(1)),\quad n\to \infty. $$
M. Ganzburg and S. Tikhonov generalized this result to the case of Nikolskii–Bernstein constants:
$$ \mathcal{C}(n;p;r)=n^{r+1/p}\mathcal{L}(p;r)(1+o(1)),\quad n\to \infty. $$
They also showed the existence of the extremal polynomial $\tilde{T}_{n,r}$ and the function $\tilde{F}_{r}$ in this problem, respectively. Earlier, we gave more precise boundaries in the Levin–Lubinsky-type result, proving that for all $p$ and $n$
$$ n^{1/p}\mathcal{L}(p;0)\le \mathcal{C}(n;p;0)\le (n+\lceil 1/p\rceil)^{1/p}\mathcal{L}(p;0). $$
Here we establish close facts for the case of Nikolskii–Bernstein constants, which also imply the asymptotic Ganzburg–Tikhonov equality. The results are stated in terms of extremal functions $\tilde{T}_{n,r}$, $\tilde{F}_{r}$ and the Taylor coefficients of a kernel of type Jackson–Fejer $(\frac{\sin \pi x}{\pi x})^{2s}$. We implicitly use Levitan-type polynomials arising from the Poisson summation formula. We formulate one hypothesis about the signs of the Taylor coefficients of the extremal functions.