Conjecture on the generators of the abelianisation of Gamma_n(1,n/2-1)

Let $\Gamma_n(1,n/2-1)$ be the cyclically presented group in the source, let $\Gamma_n(1,n/2-1)^\mathrm{ab}$ denote its abelianisation, and let $d(G)$ be the minimum number of generators of a finite abelian group $G$. Generator-number conjecture. Suppose $(n,6)=2$ and $n\geq 8$. Then

$$d(\Gamma_n(1,n/2-1)^\mathrm{ab})=\begin{cases}1&\mathrm{if}~(n,16)=2,\\2&\mathrm{if}~(n,16)=4~\mathrm{or}~16,\\3&\mathrm{if}~(n,16)=8.\end{cases}$$

The conjecture was verified computationally using GAP for $n\leq 500$; the paper notes that the third case also has an established lower bound of three, while the remaining cases are not proved there.