Exact constant in the Jackson–Stechkin inequality in the space $L^2$ on the period

In the space $L^2$ of real-valued measurable $2\pi$-periodic functions that are square summable on the period $[0,2\pi]$, the Jackson—Stechkin inequality
$$ E_n(f)\le\mathcal K_n(\delta,\omega)\omega(\delta,f),\quad f\in L^2, $$
is considered, where $E_n(f)$ is the value of the best approximation of the function $f$ by trigonometric polynomials of order at most $n$ and $\omega(\delta,f)$ is the modulus of continuity of the function f in $L^2$ of order 1 or 2. The value
$$ \mathcal K_n(\delta,\omega)=\sup\biggl\{\frac{E_n(f)}{\omega(\delta,f)}:f\in L^2\biggr\} $$
is found at the points $\delta=2\pi/m$ (where $m\in\mathbb N$) for $m\ge3n^2+2$ and $\omega=\omega_1$ as well as for $m\ge11n^4/3-1$ and $\omega=\omega_2$.