On the Bernstein's constant in convex approximation

Denoting by $E_{n}^{(+2)}(f)$ the best uniform approximation of $f$ by convex polynomials of degree $\le n$, there is an open question if there exists the limit $\lim_{n\to \infty}n^λE_{n}^{(+2)}(|x|^λ)$ for $λ\ge 1$.

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