Krasikov's stronger conjecture for Jacobi polynomial extrema

Let $P_k^{(\alpha,\beta)}(x)$ be the orthonormal Jacobi polynomial, with $k\geq 0$ and $\alpha\geq\beta\geq -\frac{1}{2}$. Define

$$M_k^{\alpha,\beta}(x)=\sqrt{1-x^2}\,(1-x)^\alpha(1+x)^\beta\left(P_k^{(\alpha,\beta)}(x)\right)^2,$$

and

$$\mathcal{M}_k^{\alpha,\beta}=\max_{x\in[-1,1]}M_k^{\alpha,\beta}(x).$$

Krasikov's stronger conjecture. One has

$$\mathcal{M}_k^{\alpha,\beta}=O\left(\max\left\{1,|\alpha|^{1/3}\left(1+\frac{|\alpha|}{k}\right)^{1/6}\right\}\right).$$

This improves the Erdélyi–Magnus–Nevai prediction in the parameter growth. The paper presents evidence for it and gives only weaker upper bounds, so the conjecture remains open.