The Grothendieck-group form of the Tachikawa conjecture

Let $k$ be a field, let $A$ be a finite-dimensional $k$-algebra, and let $n$ be the number of pairwise non-isomorphic simple $A$-modules. Let $\mathsf{D}^{\rm b}(\mathsf{mod} A)$ be the bounded derived category and $\mathsf{K}^{\rm b}(\mathsf{proj} A)$ the bounded homotopy category of finitely generated projective $A$-modules. The Grothendieck-group form of the Tachikawa conjecture. There does not exist a thick subcategory ${\mathcal T}$ of $\mathsf{D}^{\rm b}(\mathsf{mod} A)$ containing $\mathsf{K}^{\rm b}(\mathsf{proj} A)$ such that the Grothendieck group $K_0({\mathcal T})$ is a free abelian group with rank strictly bigger than $n$. The source proves that this conjecture implies the silting form of the Auslander–Reiten conjecture, but gives no resolution of the conjecture itself.