Relative Brauer relation generation conjecture for finite p-groups

Let $p$ be a prime, let $G$ be a finite $p$-group, and let $K(G,C_p)$ denote the module of relative $(G,C_p)$-Brauer relations. For a sub-quotient $P=K/N$ of $G\times C_p$, write $\operatorname{Ind}^{G\times C_p}_K\operatorname{Inf}^{K}_P(\Theta'_P)$ for the relation induced and inflated from a $P$-relation $\Theta'_P$. Relative Brauer relation generation conjecture. All relative Brauer $(G,C_p)$-relations are integer linear combinations of relations of the form

$$\Theta=\sum n_P\operatorname{Ind}^{G\times C_p}_K\operatorname{Inf}^{K}_P(\Theta'_P),\qquad P=K/N\text{ a sub-quotient of }G\times C_p,$$

where each $P$ is isomorphic to one of $C_p\times C_p\times C_p$, the Heisenberg group of order $p^3$ times $C_p$, or, when $p=2$, the dihedral group of order $2^n$ with $n\geq 4$ times $C_2$. This would identify the listed sub-quotients as the sources of all relative Brauer relations for finite $p$-groups; the statement is presented as a conjectural formulation based on rank estimates and a search for suitable relations, and its resolution is not supplied here.