The global-dimension extension for trace monoid diagram categories

Let ${\mathcal A}$ be an Abelian category with coproducts, and let $M(E,I)$ be a trace monoid. The preceding theorem establishes

$${\rm gl.\dim\,} {\mathcal A}^{M(E,I)} = n + {\rm gl.\dim\,} {\mathcal A}$$

when the maximal cardinality of pairwise independent elements of $E$ is $n<\infty$ and ${\mathcal A}$ has exact coproducts or enough projectives.

Global-dimension extension. This is true for all Abelian categories with coproducts.

The assertion proposes removing both additional hypotheses from the theorem, extending the Hilbert-syzygy-type formula to arbitrary Abelian categories with coproducts. No resolution status is supplied in the source.