Let $X$ be a variety over a field $k$. We have the bounded derived category of coherent sheaves $D^b_{coh}(X)$ and the derived category of perfect complex $Perf(X)$. It is clear that $Perf(X)$ is a strictly full triangulated subcategory of $D^b_{coh}(X)$. Then following Orlov 2003 we define the triangulated category of singularities of $X$ as the quotient of $D^b_{coh}(X)$ and $Perf(X)$, i.e. $$ D_{sg}(X)=D^b_{coh}(X)/Perf(X). $$

Recall that we call $\mathcal{A}\to \mathcal{B}\to\mathcal{C}$ an exact sequence of triangulated categories if the composition sends $\mathcal{A}$ to zero, $\mathcal{A}\to \mathcal{B}$ is fully faithful and coincides (up to equivalence) with the subcategory of those objects in $\mathcal{B}$ which are zero in $\mathcal{C}$, and the induced functor $\mathcal{B}/\mathcal{A}\to \mathcal{C}$ is an equivalence. It is easy to verify that $$ Perf(X)\to D^b_{coh}(X)\to D_{sg}(X) $$ is an exact sequence of triangulated categories.

On the other hand we have the Grothendieck groups of the above triangulated categories. In more details we define $K^0(X)$ the Grothendieck groups of $Perf(X)$, $K_0(X)$ the Grothendieck groups of $ D^b_{coh}(X)$, and $K_0( D_{sg}(X))$ the Grothendieck groups of $ D_{sg}(X)$. Then from the exact sequence $Perf(X)\to D^b_{coh}(X)\to D_{sg}(X)$ we get an exact sequence of abelian groups $$ K^0(X)\to K_0(X)\to K_0(D_{sg}(X))\to 0. $$ See Schlichting 2008 Exercise 3.1.6.

We would like to extend the above exact sequence to the left, via higher algebraic K-theory. However, there is not higher K-theory on merely triangulated categories. Nevertheless we have $K^i(X)$ and $K_i(X)$ for $i\geq 1$ in the framework of complicial exact categories. $\textbf{My question}$ is: could we define the higher K-theory of $D_{sg}(X)$ and get a long exact sequence $$ \ldots \to K^i(X)\to K_i(X)\to K_i(D_{sg}(X))\to K^{i-1}(X)\to \ldots ? $$