Consider a sort of $\mathbb{A}^1$-homotopy-stable algebraic $K$-theory for rings constructed as follows.

For $K_0$ we take a symmetrization subject to natural direct sum operation of $\mathbb{A}^1$-homotopy classes of finite-dimensional idempotents in $\mathbb{M}_\infty(B)$. Higher and lower groups are defined using looping/suspension. I hypothesize that this construction coincides with Weibel's homotopy algebraic K-theory $KH$, which is originally defined via spectra. This my be a trivial fact not even worth describing in the literature, or quite possibly I'm terribly wrong and missing some simple counterexample.

(Sorry if the question is too simple or too unclear - I haven't been doing real math for years.)