Stable inversion configurations for odd numbers of weighted rays

Let $m=2k+1$, let $c_1,\ldots,c_m$ be specified weights, let $\gamma_1,\ldots,\gamma_m$ denote ray directions, and let $\mathcal{S}$ be the corresponding star transform. A configuration has a stable inversion when $\mathcal{S}$ does not contain singular directions of Type 2. Stable inversion configuration conjecture. For any set of specified weights $c_1,\ldots,c_m$, there exist $\gamma_1,\ldots,\gamma_m$ such that $\mathcal{S}$ does not contain singular directions of Type 2; equivalently, for any set of weights there are configurations of $\mathcal{S}$ with a stable inversion. This extends the established result for $m=3$ to odd $m>3$; the source states that the generalization is technically more complicated and is intended for future work.