  We analyse the spectral convergence of high order elliptic differential
operators subject to singular domain perturbations and homogeneous boundary
conditions of intermediate type. We identify sharp assumptions on the domain
perturbations improving, in the case of polyharmonic operators of higher order,
conditions known to be sharp in the case of fourth order operators. The
optimality is proved by analysing in detail a boundary homogenization problem,
which provides a smooth version of a polyharmonic Babu\v{s}ka paradox.
