Equality of compact and pseudo-perfect Fukaya subcategories for Weinstein manifolds

Determine whether, for any Weinstein symplectic manifold X, the inclusion of categories Fuk_cpt(X) ⊂ Fuk(X)^{pp} is an equality; equivalently, show that every pseudo-perfect object in the wrapped Fukaya category Fuk(X) is represented by a compact exact Lagrangian (possibly with a finite-rank local system or with the appropriate structures for immersed Lagrangians).

Background

The paper defines the subcategory Fuk_cpt(X) as the idempotent completion of compact exact Lagrangians (possibly with finite rank local systems, or immersed with Akaho–Joyce structures), and Fuk(X)^{pp} as the full subcategory of pseudo-perfect objects M for which Hom(T, M) is finite rank for all T. There is a natural inclusion Fuk_cpt(X) ⊂ Fuk(X)^{pp}.

The author notes that while Fuk(X) is known to be smooth (so pseudo-perfect modules are representable), it remains open whether every pseudo-perfect object is realized by a compact Lagrangian, i.e., whether X has “enough compact Lagrangians.”