Rescaled discrete Laguerre inequality

Let $\mathcal{L}\text{-}\mathcal{P}_n$ denote the real polynomials of degree $n$ with only real zeros, let $\mu(p)$ be their mesh size, and define

$$f_n(x,h,p)=(n-1)[p(x+h)-p(x-h)]^2-4np(x)[p(x+h)-2p(x)+p(x-h)].$$

Rescaled discrete Laguerre conjecture. If $p(x)\in\mathcal{L}\text{-}\mathcal{P}_n$ and $\mu(p)\ge h>0$, then $f_n(x,h,p)\ge 0$ for all $x\in\mathbb{R}$. This is stated as an equivalent rescaling of Krasikov's conjecture, which the paper proves; hence this formulation is solved as well.