The correctness problem for best approximations by trigonometric polynomials in the class $W_0^rH[\omega]_C$

Suppose that $k$, $r\in Z_+$, $W_0^rH[\omega]_C=\{f:f\text{ is a~$2\pi$-periodic function, }f\in C^r[-\pi,\pi],\omega(f^{(r)},\delta)\le\omega(\delta)\}$, $T_k$ is the space of trigonometric polynomials of order $k$, $p_k(f)\in T_k$ is the polynomial of best uniform approximation to $f$, and $E_k(f)$ is the error of the best approximation. It is shown that for an arbitrary $\varepsilon>0$ we have,
\begin{gather*} \sup\limits_{f\in W_0^rH[\omega]_C}\sup\limits_{\substack{q_k\in T_k\\\|f-q_k\|\le E_k(f)+\varepsilon}}\|p_k(f)-q_k\|_C\asymp R(\varepsilon), \\ \sup\limits_{f\in W_0^rH[\omega]_C}\sup\limits_{\substack{f_1\in C[-\pi,\pi]\\\|f-f_1\|\le\varepsilon}}\|p_k(f)-p_k(f_1)\|_C\asymp R(\varepsilon), \end{gather*}
where for $0<\varepsilon\le\omega(1)$, $k>0$, $R(\varepsilon)$ is the root of the equation $R=(\varepsilon'R)^{r/(2k)}\omega((\varepsilon'R)^{1/(2k)})$, and for $k=0$ or $\varepsilon>\omega(1)$ we have $R(\varepsilon)=\varepsilon$.