Conjecture on the spectrum of the Veronese-curve exponent

Let $k\geq 1$ be an integer. For a real number $\zeta$, write

$$\chi_k(\zeta)=\chi(\zeta,\zeta^2,\ldots,\zeta^k)$$

and consider the exponent $\chi_k$ restricted to points on the Veronese curve. Spectrum conjecture. The spectrum of $\chi_k$ on the Veronese curve equals $[1,\infty]$.

The preceding theorem proves that the spectrum on the Veronese curve contains $[2,\infty]$, while the conjecture concerns the remaining range beginning at $1$.