Williams's conjecture on the abelianization of the groups $H(r,n,s)$

Let $H(r,n,s)=P(r,n,r+1,s,1)$, where $P(r,n,k,s,q)$ is the cyclically presented group under consideration. Williams's conjecture. For integers $r,sgeq 1$ and $ngeq 2$ satisfying

$$r\not\equiv 0\pmod n,\qquad s\not\equiv 0\pmod n,$$

one has

$$H(r,n,s)^{\operatorname{ab}}\neq 1.$$

This conjecture asserts that these groups are never perfect under the stated congruence conditions. It is cited as Williams's Conjecture 1; the supplied text gives no resolution.