If the homotopy category is well-generated, must the $\infty$-category be presentable? 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).
 A: On any complete and cocomplete category, there is a model structure where
trivial fibrations = weak equivalences = all morphisms. The homotopy category is equivalent to a single-object groupoid. So, I think that one cannot expect the converse for model categories.
A: I don't think this is actually fully answered by HA 1.4.4.2, which says that a stable $\infty$-category is presentable if and only if it's got coproducts (hence is cocomplete) and a $\kappa$-compact generator together with a locally small homotopy category (so that it's locally small.) That last condition sounds as if it would mean the homotopy category was $\kappa$-well generated, but there's no clear implication between being $\kappa$-compact in a stable $\infty$-category and being $\kappa$-compact in its underlying triangulated category. That's because being $\kappa$-compact in the homotopy category means being in the largest class of $\kappa$-small objects that is "$\kappa$-perfect" in the sense of Neeman, and it's only $\kappa$-smallness that has any clear connection to $\kappa$-filtered colimits.
However, a nearby claim is known, and was proved via a different model by Heider in his paper Two results from Morita theory of stable module categories. Heider shows that the homotopy category of a spectral model category is well generated if and only if it's triangulated equivalent to the localization of the derived category of a small spectral category at a set of objects. The derived category of a small spectral category models an arbitrary finitely presentable stable $\infty$-category, and its localizations at small sets model arbitrary accessible localizations, thus arbitrary $\alpha$-presentable stable $\infty$-categories.
Now, I'm pretty sure it's unknown precisely which stable $\infty$-categories underlie spectral model categories, so this doesn't quite answer the original question. I think it would be possible to imitate Heider's proof in general, though. Given an $\alpha$-compact generator $\mathcal G$ of the homotopy category of a stable $\infty$-category $\mathcal T$, seen as a full additive subcategory, take a small spectral category $\mathcal E$ with homotopy coherent nerve equivalent to the full subcategory of $\mathcal T$ spanned by the objects of $\mathcal G$. Then as far as I can tell, Heider's argument goes through without change to exhibit $\mathcal T$ as (the homotopy coherent nerve of) an $\alpha$-accessible localization of $D(\mathcal E^{\mathrm{op}})$. 
