Categorifying commutative rings via stably symmetric monoidal stable ∞-categories

Construct, for every commutative ring R, a small idempotent-complete stable ∞-category equipped with a stably symmetric monoidal structure such that its Grothendieck group K_0 is isomorphic to R; equivalently, determine whether for each commutative ring R there exists a stably symmetric monoidal small stable ∞-category C with K_0(C) ≃ R.

Background

The paper proves that every spectrum is equivalent to the nonconnective K-theory of a small idempotent-complete stable ∞-category, and consequently that every abelian group can be realized as K_0(C) for some such category C.

They note that extending categorification from abelian groups to (commutative) rings requires endowing the stable ∞-category with a stably symmetric monoidal structure so that K_0 acquires a ring structure compatible with R. They indicate that the strongest partial results toward this goal appear in Levy (2023), but the general problem remains unresolved.