A well-known theorem states that a category E is a Grothendieck topos if and only if it can be embedded as a reflective subcategory of a presheaf category whose reflector preserves finite limits.
In their paper Grothendieck quasitoposes, Garner and Lack proved that a category E is a Grothendieck quasitopos (the category of sheaves for one topology that are separated for another topology) if and only if it can be embedded as a reflective subcategory of a presheaf category whose reflector has stable units (i.e. preserves all pullbacks over objects in the subcategory) and preserves monomorphisms.
Is there a characterization of when a category E can be embedded as a reflective subcategory of a presheaf category whose reflector preserves finite products? (The reflector preserving finite products is equivalent to the reflective subcategory being an exponential ideal, i.e. $Y^X$ lies in the subcategory as soon as $Y$ does.) Of course, any such category must be locally presentable and cartesian closed. Can this be extended to a set of conditions which is both necessary and sufficient?
