Billey–Burdzy–Sagan positivity conjecture for peak polynomials

Let $S$ be an admissible peak set, let $m=\max(S)$, and let $p_S(x)$ be its peak polynomial. Define the forward difference operator by $(\Delta f)(x)=f(x+1)-f(x)$, and write the Newton expansion centered at $m$ as

$$p_S(x)=\sum_{j=0}^{m}(\Delta^j p_S)(m)\binom{x-m}{j}.$$

Billey–Burdzy–Sagan's positivity conjecture. Each coefficient $(\Delta^j p_S)(m)$ is a positive integer for $1\le j\le m-1$.

This conjecture asserts positivity of the binomial-basis coefficients of peak polynomials and would give an integral, positive expansion of the enumeration formula for permutations with a prescribed peak set. The paper presents it as an earlier conjecture and seeks to derive it from a stronger root-location conjecture.