Nikolov's non-negative Chebyshev expansion conjecture for snake polynomials

Let $\mu\in C[-1,1]$ be a nonnegative majorant, and let $\omega_\mu\in\mathcal{P}_n$ be the snake polynomial associated with $\mu$, meaning that $|\omega_\mu(x)|\leq\mu(x)$ on $[-1,1]$ and that it attains alternating values at $n+1$ alternation points. Nikolov's non-negative Chebyshev expansion conjecture. If $\mu\geq0$ is a continuous even convex function on $[-1,1]$, then the snake polynomials associated with $\mu$ have a non-negative expansion in the Chebyshev polynomials of the first kind. The problem asks for classes of majorants with this positivity property; the supplied text gives no evidence that the stated class has been proved sufficient.