Krasikov's discrete Laguerre inequality for polynomials

Let $\mathcal{L}\text{-}\mathcal{P}_n$ denote the real polynomials of degree $n$ with only real zeros. For such a polynomial $p$, let $\mu(p)$ be the minimum spacing between neighboring zeros. Krasikov's conjecture. If $p(x)\in\mathcal{L}\text{-}\mathcal{P}_n$ and $\mu(p)\ge 1$, then

$$(n-1)[p(x+1)-p(x-1)]^2-4np(x)[p(x+1)-2p(x)+p(x-1)]\ge 0$$

for all $x\in\mathbb{R}$. This discrete analogue of the classical Laguerre inequality is proved in the paper, so the conjecture is solved.