Weak homotopy equivalence for convex disks with Legendrian boundary

Let $(M,cxi)$ be an overtwisted contact $3$-manifold, and let the hypotheses of the principal theorem hold. Denote by the inclusion in the main inclusion the map from the space of convex disks with Legendrian boundary to the corresponding space of embedded disks. Disk inclusion conjecture. The inclusion is a weak homotopy equivalence. The conjecture proposes extending the principal theorem from the cases established there to higher homotopy groups; the paper explains that the obstruction is adapting the isotopy construction because Giroux's genericity theorem does not extend to higher-dimensional families of embeddings.