Weak homotopy equivalence for Legendrians in a Darboux ball

Let $(M,\xi)$ be an overtwisted contact $3$-manifold, and let $\Lambda\subseteq(M,\xi)$ be an embedded Legendrian lying in some Darboux ball. Denote by the inclusion in the Legendrian inclusion the map from the Legendrian embedding space to the corresponding embedding space. Legendrian inclusion conjecture. The inclusion is a weak homotopy equivalence. This is the Legendrian analogue of the proposed result for convex disks. The paper presents it as an open question in the setting of overtwisted contact $3$-manifolds; the precise spaces involved are specified by the referenced inclusion in the full paper.