Theorem: Assume VP. Let $\mathcal{M}$ be an accessible model category such that there exists a set of generating cofibrations $I$ and such that all objects are fibrant. Then it is combinatorial.

**Proof**: Consider the left determined model structure with respect to $I$: the minimal $I$-localizer is denoted by $\mathcal{W}_I$. Denote by $\mathcal{W}$ the class of weak equivalences of $\mathcal{M}$. Then $\mathrm{cof}(I) \cap \mathcal{W}_I \subset \mathrm{cof}(I) \cap \mathcal{W}$. Therefore all objects of the left determined model structure are fibrant. Since a model structure is characterized by its class of cofibrations and its class of fibrant objects, $\mathcal{M}$ is left determined, and therefore is combinatorial.

Question: Can we remove VP in the statement of the theorem ? I mean, is it as difficult as Smith's conjecture about left determined model structures ? Note that in the proof, I don't use the fact that the model category $\mathcal{M}$ is accessible, I only use the fact that the underlying category is locally presentable. The question is: is it easier when one already knows that $\mathcal{M}$ is accessible as a model category ?