Real-root location conjecture for peak polynomials beyond the final gap

Let $S$ be an admissible peak set with $|S|\ge 2$, and let $S_1$ denote the set obtained by removing the largest element of $S$. Let $p_S(x)$ be the peak polynomial, and let $x_0\in\mathbb R$ satisfy $p_S(x_0)=0$.

Real-root location conjecture. The inequality $x_0>\max(S_1)$ holds if and only if

$$x_0=\max(S).$$

The conjecture concerns the location of real roots relative to the penultimate peak. The paper proves the assertion for all integer values of $x_0$, but leaves the case of arbitrary real $x_0$ open.