We consider general 'arrangements' in stratified spaces with tangential structure and their locally-constant factorisation algebras, the simplest of which (besides the open disk) is the half-plane arrangement, giving Voronov's Swiss-Cheese operad. We formulate analogues of the higher Deligne conjecture for such arrangements, defining stratified and structured Hochschild objects for this purpose, and give proofs in some cases. These are equivalently classification statements for constructible factorisation algebras on these spaces.