Discrete logarithmic-derivative measure conjecture

Let $\mathcal{L}\text{-}\mathcal{P}_n$ denote the real polynomials of degree $n$ with only real zeros, let $\mu(p)$ be the mesh size, and let $m$ denote Lebesgue measure. Discrete measure conjecture. If $p\in\mathcal{L}\text{-}\mathcal{P}_n$, $n\ge 2$, and $\mu(p)\ge 1$, then

$$m\left(\left\{x\in\mathbb{R}:\frac{p(x+1)-p(x)}{p(x)}\ge\lambda\right\}\right)=\frac{n}{\lambda}$$

for all $\lambda>0$. This is the discrete analogue of the corresponding theorem for the logarithmic derivative, and the supplied text does not report a resolution.