Suppose that $R\to S$ is a 1-connected morphism of connective structured ring spectra that induces an isomorphism on rational homotopy groups. Is the induced map of (Waldhausen) K-theory spectra
$$
K(R) \to K(S)
$$
also an equivalence on rational homotopy?


(The case of the map of group rings $S^0[G] \to \Bbb Z[G]$ was answered in the affirmative in one of Waldhausen's early papers.)


I am looking for a solid reference (assuming the result to be true; I believe it is).



Rational homotopy invariance of algebraic $K$-theory
The theorem can be found in more general form in Land, Tamme [On the K-theory of pullbacks](https://arxiv.org/abs/1808.05559), Lemma 2.4.


