Triangulated telescope conjecture

Let $\mathscr{T}$ be a compactly generated triangulated category with all coproducts, and let

$$j:\mathscr{T}\rightleftarrows\mathscr{T}':j_\rho$$

be a smashing localization, meaning that its right adjoint preserves coproducts. Write $\ker(j)$ for the full subcategory of $\mathscr{T}$ consisting of objects $x$ such that $j(x)\simeq 0$. Triangulated telescope conjecture. If $j$ is a smashing localization, then $\ker(j)$ is generated by objects that are compact in $\mathscr{T}$. This conjecture generalizes the telescope conjecture from stable homotopy theory to compactly generated triangulated categories; its status is not established in the supplied text, although the paper proves it in several geometric settings.