Jeff Smith's theorem gives a simple criterion for the existence of a combinatorial model category. Is there a similar theorem for combinatorial left semi-model categories? I see two problems that occur when you try to apply the theorem in this context.

First, I don't think that the class of weak equivalences is necessarily an accessibly embedded accessible full subcategory of the category of morphisms. We can try to fix this by modifying this condition so that it says something about weak equivalences with cofibrant domains only, but I don't know what this modification might be.

The second problem I see is that we cannot prove that $\mathrm{inj}(J) \cap \mathcal{W} \subseteq \mathrm{inj}(I)$. Usually, this follows from the fact that $\mathrm{cof}(I) \cap \mathcal{W} \subseteq \mathrm{cof}(J)$, but in a semi-model category the latter condition only holds for maps with cofibrant domains while the former condition should be true for all maps.