Suppose $ \mathcal{C}$ is a stable $ \infty$ -category whose homotopy category is a well-generated triangulated category in the sense of Neeman’s book. Must $ \mathcal{C}$ be a presentable $ \infty$ -category?

Rosicky proved that if $ \mathcal{M}$ is a combinatorial stable model category then $ Ho(\mathcal{M})$ is well-generated, so I’m asking for a kind of converse, and I’m happy for an answer either about model categories being combinatorial or about $ \infty$ -categories being presentable.

The starting point is probably trying to figure out whether or not the isomorphisms of $ Ho(\mathcal{M})$ form an accessible subcategory of the arrow category, which seems like something you’d need (since in order for $ \mathcal{M}$ or $ \mathcal{C}$ to be presentable you need to know the weak equivalences are an accessible subcategory of the arrow category, see A.2.6.6 of Lurie’s HTT). It feels like the sort of thing that should be true, but I also don’t know if it’s enough to then deduce $ \mathcal{C}$ is presentable (though Lurie’s A.2.6.9 is surely relevant).