Truncated binomial polynomial conjecture for boundary points of Steiner cones

Let $n\geq 5$ and let $\gamma\in\mathop\mathrm{bd}\nolimits\mathcal{R}(n)\backslash\mathbb{R}_{\leq 0}$. For $0\leq j<k\leq n$, define the truncated binomial polynomial

$$P_{j,k}^n(z):=\sum_{i=j}^k\binom{n}{i}z^i.$$

Truncated binomial polynomial conjecture. There exist a truncated binomial polynomial $P_{j,k}^n(z)$ with $0<j<k<n$ and a real number $\lambda>0$ such that

$$P_{j,k}^n(\lambda\gamma)=0.$$

The cones $\mathcal{R}(n)$ encode the roots of Steiner polynomials, and the conjecture proposes that every boundary point outside the nonpositive real axis is, up to positive scaling, a root of a suitably truncated binomial polynomial. The preceding results establish the corresponding description in dimensions $3$ and $4$, while the assertion is proposed here for $n\geq 5$.