Tensor telescope conjecture

Let $\mathscr{T}$ be a compactly generated $\otimes$-triangulated category and let $\mathscr{U}$ be a compactly generated triangulated category, both with all coproducts. Suppose

$$\otimes:\mathscr{T}\times\mathscr{U}\rightarrow\mathscr{U}$$

preserves coproducts in each variable, and call a localizing subcategory of $\mathscr{U}$ $\mathscr{T}$-closed when it is closed under tensor products with objects of $\mathscr{T}$. Tensor telescope conjecture. If $j:\mathscr{U}\rightleftarrows\mathscr{U}':j_\rho$ is a smashing localization whose kernel is $\mathscr{T}$-closed, then $\ker(j)$ is generated by objects that are compact in $\mathscr{U}$. This is the tensor-action version of the telescope conjecture; the supplied text gives no resolution in full generality, while the paper establishes it for important classes of geometric categories.