Finite-cohomology generation conjecture for homotopy categories of injectives

Let $G$ be a finite group and let $\mathsf{k}$ be a field of characteristic $p$. Let $\mathsf{KInj}(\mathsf{k}G)$ be the homotopy category of complexes of injective $\mathsf{k}G$-modules. Let $\mathsf{KInj}^{b}(\mathsf{k}G)$ be the thick subcategory of objects $X$ for which $H^*(G,X)$ is finitely generated over $H^*(G,\mathsf{k})$, and let $\mathsf{KInj}^{0}(\mathsf{k}G)$ be the localising subcategory of objects with $H^*(G,X)=0$. The bounded derived category $\mathsf{D}^{b}(\mathsf{k}G)$ is identified with the compact objects in $\mathsf{KInj}(\mathsf{k}G)$.

Finite-cohomology generation conjecture.

$$\mathsf{KInj}^{b}(\mathsf{k}G)=\mathsf{Thick}(\mathsf{D}^{b}(\mathsf{k}G),\mathsf{KInj}^{0}(\mathsf{k}G)).$$

Thus every object with finitely generated cohomology should be generated by bounded-derived objects and no-cohomology objects. The paper proves this conjecture is equivalent to the corresponding stable-module-category conjecture and records that the general weaker conjecture remains open.