Jackson inequality for polynomials on a circular arc

For $T>1$, let $A$ be the open arc of the complex unit circle

$$A=\left\{e^{i\theta}:\theta\in\left(-\frac{\pi}{T},\frac{\pi}{T}\right)\right\},$$

and define

$$\mu(z)=\sqrt{(z-e^{\frac{i\pi}{T}})(z-e^{-\frac{i\pi}{T}})}.$$

For $F\in C^1(A)$, write $F'(z)$ for its derivative along the arc. Jackson inequality for polynomials on a circular arc. There exists a constant $C_T>0$ such that, for every $n\in\mathbb{N}$ and every $F\in C^1(A)$, there is a polynomial $q_n$ of degree $n$ satisfying

$$\sup_{z\in A}\left|F(z)-q_n(z)\right|\leq\frac{C_T}{n}\sup_{z\in A}\left|\mu(z)F'(z)\right|.$$

This would provide the Jackson-type approximation estimate needed to derive convergence rates for Fourier extensions on a circular arc; the supplied text gives no evidence that the inequality has been proved or disproved.