Bukh's conjecture on spherical codes with separated negative inner products

Let $\beta\in(0,1)$ be fixed, let $\alpha_1,\dots,\alpha_k$ be any $k$ real numbers, and let a spherical $L$-code be a finite non-empty set of unit vectors in $\mathbb{R}^n$ whose pairwise inner products lie in $L$. Bukh's conjecture. Every spherical $[-1,-\beta]\cup\{\alpha_1,\dots,\alpha_k\}$-code in $\mathbb{R}^n$ has size at most

$$c_{\beta,k}n^k,$$

for some constant $c_{\beta,k}$ depending only on $\beta$ and $k$. Bukh had proved the corresponding linear bound when the positive part consists of one value. The conjecture generalizes polynomial upper bounds of Delsarte, Goethals and Seidel and remains open based on the supplied information.