Polynomial bound for spherical codes with a negative interval and k positive inner products

Let $N_L(d)$ denote the maximum size of a spherical code in $\mathbb{R}^d$ whose pairwise inner products belong to $L$. For $\beta$ and $k$ as in the conjecture, and for any real numbers $\alpha_1,\dotsc,\alpha_k$, set

$$L=[-1,-\beta]\cup\{\alpha_1,\dotsc,\alpha_k\}.$$

Polynomial spherical-code conjecture. Suppose $\alpha_1,\dotsc,\alpha_k$ are any $k$ real numbers. Then

$$N_L(d)\leq c_{\beta,k}d^k.$$

This extends the linear bound for a single allowed inner product and the constant bound for inner products in $[-1,-\beta]$. The conjectured degree-$k$ estimate is left open in the source.