Pruned arboreal singularities and loose Legendrians

Originally defined by Nadler, an arboreal singularity of a Lagrangian is a robust kind of singularity; notable in that any Lagrangian singularity can be deformed into one with only arboreal singularities. They are also interesting algebraically, because an arboreal singularity is determined by a quiver (a rooted tree), and the space of constructable sheaves on the arboreal singularity coincides with the space of representations of the quiver. However, often times an arboreal Lagrangian skeleton will have free boundary components, so the space of sheaves is considerable smaller, even locally. Geometrically this corresponds to removing a number of the top dimensional strata, which we call pruning. We prove that the link of a pruned arboreal singularity is loose, if and only if the singularity admits no non-constant constructable sheaves. We'll also discuss how this fits into a larger program of using the wrapped Fukaya category to detect flexibility of Weinstein manifol