Károlyi's inverse Erdős–Heilbronn conjecture for non-nilpotent groups

Let $A,B$ be nonempty subsets of a finite, not necessarily abelian, non-nilpotent group $G$, and let $p(G)$ be the quantity defined in the source as the minimal torsion element. Put $|A|=k\geq 3$ and $|B|=l\geq 3$, with $k+l-3<p(G)$. Define the restricted product set

$$A\overset{\iota}{\cdot}B=\{ab\mid a\in A,\ b\in B,\ a\neq b\}.$$

Károlyi's inverse Erdős–Heilbronn conjecture. The equality

$$|A\overset{\iota}{\cdot}B|=|A|+|B|-3$$

holds if and only if there exist $a,q\in G$ such that

$$A=\{a,aq,aq^2,\dots,aq^{k-1}\},\qquad B=\{a,qa,q^2a,\dots,q^{l-1}a\},$$

where $aq^{k-1}=q^{l-1}a$, so that $A$ and $B$ share the same endpoints. The conjecture seeks the precise inverse characterization of equality in the non-nilpotent setting; the source notes preceding results for abelian groups, asymptotic results, and finite nilpotent groups, but does not report a proof in the stated generality.