Formality of a higher-codimensional Swiss-cheese operad

We study bicolored configurations of points in the Euclidean n-space that are constrained to remain either inside or outside a fixed Euclidean m-subspace, with n - m >= 2. We define a higher-codimensional variant of the Swiss-cheese operad, called the complementarily constrained disks operad CDm n, associated to such configurations. The operad CDm n is weakly equivalent to the operad of locally constant factorization algebras on the stratified space {R-m subset of R-n}. We prove that this operad is formal over R.