Categorical explanation for the E_infinity-algebra structure on a K-theoretic cofiber

Identify and explain the categorical mechanism that endows the cofiber of the map K(C_{<1}) → K(C) with a natural K(C)-E_infinity-algebra structure, where C is an algebraically closed nonarchimedean field of residue characteristic p and C_{<1} denotes the non-unital Tor-unital subring of elements of norm < 1.

Background

In the K-theory section, the authors identify a description of the functor \overline{K} in terms of the cofiber of K(C_{<1}) → K(C) for an algebraically closed nonarchimedean field C. From this they observe a natural K(C)-E_infinity-algebra structure on the cofiber, an outcome they find unexpected.

They explicitly state that the categorical reasons for this E_infinity structure are not clear, indicating an unresolved conceptual explanation for this phenomenon.