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 ${\lim }_{n\to \mathrm{\infty }}{n}^{\lambda }{E}_n^{(+2)}(\mathrm{\mid }x{\mathrm{\mid }}^{\lambda })$ for $\lambda \ge 1$.