The prime-orbit decomposition conjecture for vanishing sums of roots of unity

Let $\mu_{2k+1}$ denote the set of $(2k+1)$-th roots of unity. For every prime factor $p$ of $2k+1$ and $\beta_p\in\left[0:\frac{2k+1}{p}-1\right]$, let

$$O_{p,\beta_p}:=\left\{\zeta^{\beta_p},\zeta^{\frac{2k+1}{p}+\beta_p},\zeta^{\frac{2(2k+1)}{p}+\beta_p},\dots,\zeta^{\frac{(p-1)(2k+1)}{p}+\beta_p}\right\}$$

be the corresponding $p$-orbit of $(2k+1)$-th roots of unity, whose elements sum to zero. Prime-orbit decomposition conjecture. Any subset of $\mu_{2k+1}$ with sum zero is a disjoint union

$$\amalg_j O_{p_j,\beta_{p_j}},$$

where each $O_{p_j,\beta_{p_j}}$ is a $p_j$-orbit and $p_j$ is a prime factor of $2k+1$. The claim proposes a general structural description of vanishing subsets of roots of unity; the supplied text gives no evidence that it has been proved or disproved, so its status remains open.