Zero-spacing preservation under a unit difference

Let $\mathcal{L}\text{-}\mathcal{P}_n$ denote the real polynomials of degree $n$ with only real zeros, and let $\mu(p)$ be the minimum spacing between neighboring zeros. For $p$, define $g(x)=p(x+1)-p(x)$. Zero-spacing conjecture. If $p(x)\in\mathcal{L}\text{-}\mathcal{P}_n$, $n\ge 2$, and $\mu(p)\ge d\ge 1$, then $\mu(g)\ge d$. This would extend the known monotonicity of mesh under differentiation to a discrete difference operator; its resolution is not given in the supplied text.