Refinement of Bernstein–Nikolskii constant for the sphere with Dunkl weight in the case of octahedron group

We continue the study of the sharp Bernstein–Nikolskii constants for spherical polynomials in the space $L^{p}(\mathbb{S}^{d})$ with the Dunkl weight. We consider the model case of the octahedral reflection group $\mathbb{Z}_{2}^{d+1}$ and weight $\prod_{j=1}^{d+1}|x_{j}|^{2\kappa_{j}} $ when the explicit form of the Dunkl intertwining operator is known. We show that for $\min \kappa=0$ the multidimensional problem is reduced to the one-dimensional problem for the Gegenbauer weight, otherwise not.