Higher-order discrete Laguerre inequality for real-rooted polynomials

Let $p$ be a real-rooted polynomial of degree $k$, and let $M(p)$ denote its mesh, with $M(p)\geq 1$. Discrete Laguerre inequality. Then

$$(k-1)(p(x+1)-p(x-1))^2-4k p(x)(p(x+1)-2p(x)+p(x-1))\geq 0.$$

This conjecture proposes a discrete analogue of a sharper polynomial Laguerre inequality, extending the known first-order discrete result to an expression involving second differences. The source does not state whether it has been proved or disproved.