Let $R$ be a ring and let $\mathcal{C}$ be the category of perfect $R$-complexes. Suppose that $$S=\bigoplus_{i=1}^{\infty}R$$

Let us define $\mathcal{D}$ the smallest thick category generated by $S$.

Clearly $\mathcal{C}$ is a full subcategory of $\mathcal{D}$

The natural embedding $i:\mathcal{C}\rightarrow \mathcal{D} $ induces a morphism in algebraic $K$-theory given by $K(i):K(\mathcal{C})\rightarrow K(\mathcal{D}) $

My question is the following: What can be said about $K_{n}(\mathcal{C})\rightarrow K_{n}(\mathcal{D}) $ for each $n\in \mathbf{N}$, is it an injective/surjective homomorphism?