This question is related to the MO questions What is the difference between Grothendieck groups K_0(X) vs K^0(X) on schemes? and Does a fully faithful functor between triangulated categories induce embedding of their Grothendieck groups?

Let $X$ be a noetherian scheme over a field $k$. We have the derived category of perfect complexes on $X$, $Perf(X)$, and the derived category of bounded complex of coherent sheaves on $X$, $D^b(coh(X))$. It is clear that $Perf(X)$ is a strictly full triangulated subcategory of $D^b(coh(X))$ and for non-regular $X$, these two categories are not equivalent.

Now we define $K^0(X)$ be the Grothendieck group of $Perf(X)$ and $K_0(X)$ be the Grothendieck group of $D^b(coh(X))$. Then the embedding $Perf(X)\to D^b(coh(X))$ induces a homomorphism $$ K^0(X)\to K_0(X). $$

$\textbf{My question}$ is: is the above homomorphism an momomorphism? (under some assumptions on $X$ or the base field $k$, if necessary) If not, is there any counter-example?