Let $X$ be a smooth projective variety and write $\mathbf{D}(X)$ for its triangulated category of perfect complexes of quasi-coherent sheaves. Recall that $\mathbf{D}(X)$ determines the Grothendieck group $K_0(X)$. Therefore by the Grothendieck-Riemann-Roch theorem, $\mathbf{D}(X)$ also determines the (direct sum of the) Chow groups $\mathrm{CH}_*(X, \mathbf{Q})$ with rational coefficients.

Are there known counterexamples to this statement for integral coefficients?

(I should note that I consider $\mathbf{D}(X)$ only as a triangulated category, in particular without its monoidal structure.)