Many first courses on Floer homology, which I have seen, use the question of nondisplaceability as a motivation. People usually begin with saying that they want to study the question whether or not, in a compact symplectic manifold $X$, one can displace two Lagrangians $L$ and $K$ by moving one of them using a Hamiltonian isotopy. Then they would define the Floer homology $HF^*(L,K)$ by a chain complex generated by intersection points of $L$ and $K$ (after a generic perturbation) with the differential given by pseudo-holomorphic disk counting. Finally, they will explain that the homology $HF^*(L,K)$ is invariant under perturbing either $L$ or $K$ via Hamiltonian isotopies and thus we know that if $HF^*(L,K) \neq 0$, then $L$ and $K$ are nondisplaceable from each other.
For a Liouville manifold $X$, the "correct" version of Floer homology for not necessarily compact Lagrangians requires wrappings, a filtered limit over positive isotopies $$HW^*(L,K) = \varinjlim_{L^w \rightarrow L} HF^*(L^w,K)$$ where $L^w$ runs over positive isotopies of $L$ and the transition maps are given by the continuation maps. (See, e.g., arXiv:0712.3177 or arXiv:1706.03152 for details.) One reason why wrappings are necessary, which I know aside from homological mirror symmetry, is that this is necessary if one wants $HW^*$ to keep the Hamiltonian invariant property. For example, in the cotangent bundle $T^* S^1$, one can displace a fiber $T^*_x S^1$ from itself so $HF^*(T^*_x S^1) = 0$ while the wrappings will force $HW^*(T^*_x S^1) = k^{\oplus \mathbb{Z}}$.
However, this latter picture seems to contract the usual story of proving nondisplaceability by Floer homology. My guess is that this indicates that the naive nondisplaceability question is not interesting in the open setting, and one should instead consider some more sophisticated version of it by incorporating wrappings. My question is if this is already studied in the literature? Or if there are other naive symplectic topology concerns which would lead to the discovery of wrapped Floer homology?