Bounded-roots conjecture for peak polynomials

Let $S$ be an admissible peak set, let $m=\max(S)$, and let $p_S(z)$ be its peak polynomial. A complex root is a number $z\in\mathbb C$ such that $p_S(z)=0$.

Bounded-roots conjecture. Every complex root of $p_S(z)$ lies in

$$\{z\in\mathbb C: |z|\le m\text{ and }\operatorname{Re}(z)\ge -3\}.$$

The conjecture is stronger than the positivity conjecture above and is presented as a proposed explanation for positivity of the binomial-basis coefficients. The paper reports experimental verification for all admissible peak sets with $\max(S)\le 15$, but does not prove the conjecture in general.