The shifted bicyclic-unit free-product conjecture

Let $G$ be a finite group, let $H\leq G$ be a finite subgroup such that $\operatorname{Cir}_{\mathbb{Q}}^G(H)$ is non-empty, and let $g\in G$ satisfy $H\cap H^g=1$. Let $\operatorname{sb}_{H,g}$ and $\operatorname{sb}_{g^{-1},H}$ denote the shifted bicyclic maps appearing in the source, and let $\ast$ be the canonical involution of $\mathbb{Q}G$ defined by $g\mapsto g^{-1}$. The shifted bicyclic-unit conjecture.

$$\left\langle\operatorname{im}(\operatorname{sb}_{H,g}),\operatorname{im}(\operatorname{sb}_{g^{-1},H})\right\rangle\cong H\ast H\cong\left\langle\operatorname{im}(\operatorname{sb}_{H,g}),\operatorname{im}(\operatorname{sb}_{H,g})^{\ast}\right\rangle.$$

The conjecture predicts that appropriately shifted bicyclic maps realize two copies of $H$ as a free product; the source notes positive results in several nilpotent cases, while the general statement remains open.