Truncations of the ring of number-theoretic functions, revisited

Let $K$ be a field containing $\mathbb{Q}$, let $Γ$ be the ring of all functions from the positive integers to $K$ under Dirichlet convolution, and let $Γ_n$ be its truncation to functions supported on $[1,n]$. In [Snellman, Homology Homotopy Appl. 2 (2000), 17-27; arXiv:math/9904143] it was shown t