Conjecture on exclusion from the sets S^{TFFF}(n) and S^{FFFF}(n)

Let $S^{TFFF}(n)$ and $S^{FFFF}(n)$ be the classes of cyclically presented groups defined in the source, and let $\Gamma_n(1,n/2-1)$ be the corresponding group. Exclusion conjecture. Let $n\geq 19$.

1. If $n\equiv 0\pmod 6$, then

$$\Gamma_n(1,n/2-1)\notin S^{TFFF}(n).$$

2. If $n\equiv 2,4\pmod 6$, then

$$\Gamma_n(1,n/2-1)\notin S^{FFFF}(n).$$

The conjecture is based on GAP experiments comparing abelianisations for $n\leq 500$ and is stated as an answer to whether the group can belong to the indicated classes; no proof or resolution is given.