Symmetry conjecture for vectors working with the all-ones-complement vector

Let $n\in\mathbb{N}$ be odd and not divisible by $7$. Let $\mathantt{e}=(0,1,\dots,1)\in\mathbb{F}_2^n$, and let vectors work together when they satisfy the paper's cyclic covering condition. A vector $v\in\mathbb{F}_2^n$ is symmetric when it has the form $(v_0,v_1,v_2,\dots,v_2,v_1)$, equivalently $v_i=v_j$ whenever $i\equiv-j\pmod n$.

Symmetry conjecture. If $v$ and $\mathantt{e}$ work together, then $v$ is symmetric.

The conjecture is motivated by computer experiments and would imply the paper's main theorem when combined with the stated lemma excluding symmetric vectors from independent working triples. The authors explicitly say that they were unable to prove it.