The relative Burnside-kernel generation conjecture

Let $p$ be a prime, let $G$ be a finite $p$-group, and let $H\cong \mathbb Z_p$. Write $\tilde G=G\times H$, let $N(G,H)$ denote the relative Burnside kernel, and for a subquotient $L/C$ of $\tilde G$ let $\tilde N(L/C)$ be the intersection of $N(L/C)$ with the submodule $\tilde A(L/C)$ that maps into $A(G,H)$ under induction. Relative Burnside-kernel generation conjecture.

$$N(G,H)=\sum L/C\mathbin{\uparrow}\tilde N(L/C),$$

where the sum is taken over subquotients $L/C$ of $\tilde G$ isomorphic to $T\times H$, with $T$ equal to the elementary abelian group $\mathbb Z_p\times\mathbb Z_p$, the dihedral group, or the nonabelian group of order $p^3$ and exponent $p$. The source presents this assertion in the final remarks as a conjectural description of the relative kernel, and gives no resolution evidence, so its status remains open.