Integral and classified real-root conjecture for peak polynomials

Let $S=\{i_1<i_2<\cdots<i_s\}$ be an admissible peak set, and let $p_S(x)$ be its peak polynomial. The roots of $p_S(x)$ are the complex numbers $x$ satisfying $p_S(x)=0$.

Integral and classified real-root conjecture. If all roots of $p_S(x)$ are real, then all of them are integral. Moreover, $p_S(x)$ has only real roots if and only if

$$S=\{2\},\quad S=\{2,4\},\quad S=\{3\},\quad S=\{3,5\},$$

or

$$S=\{i_1<i_2<\cdots<i_s<i_s+3\},$$

or

$$S=\{i_1<i_2<\cdots<i_s<i_s+3<i_s+5\}.$$

The conjecture gives both an integrality prediction for real roots and a complete proposed classification of admissible peak sets whose peak polynomials are real-rooted. The paper says that the listed families have been shown to have only integral roots, while the converse direction remains unproved.