Agreement of Bass and Blumberg–Gepner–Tabuada non-connective K-theory

Let $\mathcal{CSA}$ be the category of connective $\mathbb{S}$-algebras, let $K(A)$ denote the connective $K$-theory functor, and let $F_n(A)$ be the functors constructed in the paper. Define the potentially non-connective functor

$$K^B(A)=\operatorname{colim}_n \Sigma^{-n}F_n(A).$$

Agreement conjecture. The non-connective $K$-theory functor $K^B$ agrees with the non-connective $K$-theory functor of Blumberg, Gepner, and Tabuada.

The conjecture identifies the delooping constructed from the Fundamental Theorem with the established non-connective $K$-theory construction. The supplied text gives no resolution, so the conjecture remains open.