Equality of the log mixed-motive categories

Let $({\rm {LM}}\flat)$ and $({\rm {LM}}\flat*)$ be the categories with the same objects and with morphisms defined using, respectively, the Chern class constructions from $K_0$ and from homotopy $K$-theory $KH$. Let $({\rm {MM}})$ and $({\rm {MM}}*)$ be the corresponding smallest full subcategories of contravariant functors generated by the functors $H^n(T)(s)$ and closed under kernels of morphisms. Equality conjecture. One has

$$({\rm {LM}}\flat)=({\rm {LM}}\flat*)\quad\text{and}\quad ({\rm {MM}})=({\rm {MM}}*).$$

The equality would show that enlarging the morphisms using homotopy $K$-theory does not change either the log-motive category or the resulting category of mixed motives. The source does not provide evidence resolving this claim.