The exact Stechkin constant conjecture at the critical uniform-metric scale

Let $C$ be the class of continuous $2\pi$-periodic functions, let $E_{n-1}(f)$ denote the error of best uniform approximation of $f$ by trigonometric polynomials of degree at most $n-1$, and let $\omega_r(f,\delta)$ be the modulus of smoothness of order $r$. Define the Stechkin constant by

$$K_{n,r}(\delta)=\sup_{f\in C}\frac{E_{n-1}(f)}{\omega_r(f,\delta)}$$

and set

$$\gamma_r^*=\frac{1}{{r\choose\lfloor r/2\rfloor}}.$$

The exact Stechkin constant conjecture. For every $r\in\mathbb{N}$,

$$\sup_{n\in\mathbb{N}}K_{n,r}\left(\frac{\pi}{n}\right):=\sup_{n\in\mathbb{N}}\sup_{f\in C}\frac{E_{n-1}(f)}{\omega_r\left(f,\frac{\pi}{n}\right)}=\gamma_r^*.$$

The conjecture asserts that the extra factor of order $\sqrt r\log r$ in the known upper bound at the critical value $\delta=\pi/n$ is not necessary. The paper establishes matching exponential-order estimates at $\delta=2\pi/n$ and an upper bound of order $\sqrt r\log r\,\gamma_r^*$ at $\delta=\pi/n$, but does not resolve the exact critical constant.