Stratified Polymorphic Typability

Let $\vdash_n$ denote the order-$n$ stratum of the polymorphic type-assignment system described in the source, obtained by restricting universal-quantifier elimination according to the level of the quantified variable in the instantiating type. For every integer $n\geq 2$, characterize the set

$$\{M : \text{there exist }\Gamma,\sigma\text{ with }\Gamma\vdash_n M:\sigma\}.$$