The perfect Prishchepov group type conjecture

Let $P(r,n,k,s,q)$ denote the Prishchepov group with parameters $r,n,k,s,q$, and let a group be perfect when its abelianization is trivial. A group is of type $\widetilde{\mathfrak{Z}}$ when it satisfies the type $\widetilde{\mathfrak{Z}}$ condition defined for Prishchepov groups. Let $n,k,q \geq 1$ and $2 \leq r<n$ with

$$\gcd(n,k-1,q)=1.$$

Suppose

$$k\not\equiv 1 \pmod{n}\qquad\text{and}\qquad k\not\equiv 1+q \pmod{n}.$$

The perfect Prishchepov group type conjecture. If $P(r,n,k,r-1,q)$ is perfect, then it is of type $\widetilde{\mathfrak{Z}}$. This conjecture predicts that every perfect nontrivial group in this parameter range has the indicated structural type. The claim is motivated by the fact that all known perfect nontrivial examples satisfying the hypotheses have this type; its general validity remains open.