Equality of compact vs. pseudoperfect Fukaya subcategories

Determine whether, for any Weinstein symplectic manifold X, the inclusion Fuk_cpt(X) ⊂ Fuk(X)^{pp} is an equality, where Fuk_cpt(X) denotes the idempotent completion of the subcategory generated by compact exact Lagrangians (with local systems) and Fuk(X)^{pp} denotes the subcategory of pseudoperfect objects for which Hom(T, M) is finite rank for all T in Fuk(X).

Background

The paper introduces the inclusion of the compact-object-generated subcategory Fuk_cpt(X) into the pseudoperfect subcategory Fuk(X)^{pp} for a Weinstein symplectic manifold X and discusses the expectation that these coincide. This equality would mean that every pseudoperfect object is represented by a compact exact Lagrangian (with appropriate brane data). Establishing this would confirm that X has 'enough compact Lagrangians' in the authors’ terminology.

This question is central for transferring noncompact mirror symmetry results to compact settings, since compact Lagrangians are the natural objects in Fukaya categories of compact symplectic manifolds obtained by compactifying X.