Let $S$ be a ring with involution (with 2 invertible). Suppose that the non connective algebraic K-theory $K(S)$ is 0 (i.e. $K_{n}(S)=0$, for all $n$).

Can we say something about the (higher) Witt groups $W_{n}(S)$ ?

Edit: I was wondering if this kind of questions is tractable in general? What kind of articles or books can be recommended to have some insight about these questions? I will be grateful to any indication. Thank you in advance for any help in this direction.

Motivation: I came to this question after reading a paper about the universal localization property of non connective algebraic $K$-theory. So the question can be formulated as follows is a pretriangulated category (with involution) has a vanishing $K$-theory does all other localizing functors such as Witt spectrum vanishes.