Conjecture 6 on centralizer-like subgroups for the 1-Engel word

Let $G \times H$ be a group, let $(u,v) \in G \times H$, and let $(u,v)^{G \times H}$ denote the normal closure of $(u,v)$ in $G \times H$. The associated subgroups $E_1^*(G \times H,(u,v))$, $R_1(G \times H,(u,v))$, and $^*E_1(G \times H,(u,v))$ are then considered. Conjecture 6.

$$E_1^*(G \times H,(u,v))=R_1(G \times H,(u,v)),$$

and

$$E_1^*(G \times H,(u,v))\triangleleft G \times H \quad\text{and}\quad E_1^*(G \times H,(u,v))\subseteq {}^*E_1(G \times H,(u,v)).$$

This conjecture is part of the authors' continuing research on counterexamples and general proofs concerning centralizer-like subgroups associated with Engel words in direct products; the supplied text does not establish whether it is open, solved, or refuted.