Krause's ideal-theoretic reformulation of the telescope conjecture

Let $\mathcal{D}$ be a compactly generated triangulated category, and let $\mathbf{t}=(\mathcal{X},\mathcal{Y})$ be a smashing semiorthogonal decomposition of $\mathcal{D}$. Let $\mathcal{D}^c$ denote the compact objects, and let $I_{\mathbf{t}}$ be the ideal of $\mathcal{D}^c$ consisting of morphisms $\alpha\colon C\to C'$ such that

$$\mathcal{D}(\alpha,Y)\colon\mathcal{D}(C',Y)\to\mathcal{D}(C,Y)$$

is zero for every $Y\in\mathcal{Y}$.

Krause's reformulated telescope conjecture. The ideal $I_{\mathbf{t}}$ is generated by identity morphisms of objects in $\mathcal{D}^c$.

Krause's reformulation is equivalent to the compact-generation formulation because the decomposition is compactly generated precisely when its associated ideal is generated by such identities. It is refuted in the same generality as the telescope conjecture.