Steiner polynomial root-location conjecture

Let $K,E\in\mathcal{K}^n$ be convex bodies, and let

$$f(K,E,s)=\sum_{i=0}^n\binom{n}{i}\mathrm{W}_i(K;E)s^i$$

be their relative Steiner polynomial. Let $a_1\leq\dots\leq a_n$ be the real parts of the roots of $f(K,E,s)$, and let $\mathrm{r}(K;E)$ and $\mathrm{R}(K;E)$ denote the relative inradius and circumradius. Steiner polynomial root-location conjecture. If $a_1\leq\dots\leq a_n$ are the real parts of the roots of $f(K,E,s)$, then

$$a_1\leq-\mathrm{R}(K;E)\leq-\mathrm{r}(K;E)\leq a_n\leq0.$$

This conjecture extends the planar root-location property and arises from Teissier's problem concerning higher-dimensional analogues of Bonnesen-type inequalities for intersection numbers. It was posed by Shephard and Yao and remains unresolved in general.