The generic bicyclic-unit free-group conjecture

Let $G$ be a finite group that is not Dedekind, let $R$ be the coefficient ring, and let $\operatorname{Bic}_R(G)$ be the group generated by all bicyclic units of $RG$. Fix a bicyclic unit $\beta$. The generic bicyclic-unit free-group conjecture. The set

$$\left\{\alpha\in\operatorname{Bic}_R(G)\mid \langle\alpha,\beta\rangle\cong\langle\alpha\rangle\ast\langle\beta\rangle\ \text{canonically}\right\}$$

is “large” in $\operatorname{Bic}_R(G)$. The conjecture makes precise the expectation that two bicyclic units generically generate a free product, but the source leaves “large” unspecified; its precise formulation and general validity remain open.