**Question:** Let $\mathcal A$ be an abelian category. Suppose that the only Serre subcategories of $\mathcal A$ are the zero category and $\mathcal A$ itself. Does it follow that every short exact sequence splits in $\mathcal A$?

**Notes:**

 - The hypothesis of no Serre subcategories means that the locally coherent Grothendieck category $Ind(\mathcal A)$ has trivial Ziegler spectrum.

 - The conclusion that short exact sequences in $\mathcal A$ split is [not quite as strong](https://mathoverflow.net/questions/327899/name-for-abelian-category-in-which-every-short-exact-sequence-splits) as saying that $A$ is semisimple.

 - If $0 \to A \to B \to C \to 0$ is a non-split short exact sequence, then there is at least a weak Serre class given by $\{X \mid Ext^\ast(C,X) = 0\}$ which is not all of $\mathcal A$ (though maybe it could still be zero). But I'm not sure how to get an actual Serre class out of this.