Sharp Jackson–Stechkin inequality in $L^2$ for multidimensional spheres

In this paper we prove the Jackson–Stechkin inequality
$$ E_{n-1}(f)<\omega_r(f,2\tau_{n,\lambda}), \qquad n\ge1, \quad m\ge5, \quad r\ge1, $$
$f\in L^2(\mathbb S^{m-1})$, $f\not\equiv\textrm{const}$, which is sharp for each $n=2,3,\dots$; here $E_{n-1}(f)$ is the best approximation of a function $f$ by spherical polynomials of degree $\le n-1$, $\omega_r(f,\tau)$ is the $r$th modulus of continuity of $f$ based on the translations
$$ s_tf(x)=\frac 1{|\mathbb S^{m-2}|}\int_{\mathbb S^{m-2}}f(x\cos t+\xi\sin t)\,d\xi, \qquad t\in\mathbb R, \quad x\in\mathbb S^{m-1}, $$
$\mathbb S^{m-2}=\mathbb S^{m-2}_x=\bigl\{\xi\in \mathbb S^{m-1}:x\cdot\xi=0\bigr\}$, $|\mathbb S^{m-2}|$ is the measure of the unit Euclidean sphere $\mathbb S^{m-2}$, $\lambda=(m-2)/2$ and $\tau_{n,\lambda}$ is the first positive zero of the Gegenbauer cosine polynomial $C^\lambda_n(\cos t)$ .