The telescope conjecture without smash product

Let $\mathcal{T}$ be a rigidly compactly generated tensor triangulated category. A localization $L:\mathcal{T}\to\mathcal{T}$ is smashing if it preserves coproducts, and it is finite if $\operatorname{Ker}L$ is generated by a set of compact objects of $\mathcal{T}$. The telescope conjecture without smash product. Every smashing localization

$$L:\mathcal{T}\to\mathcal{T}$$

is finite. This is the tensor-free formulation of the telescope conjecture; it is known to be false in this generality, while it agrees with Ravenel's telescope conjecture for spectra and, more generally, for rigidly compactly generated tensor triangulated categories.