  The third named author has been developing a theory of "higher" symplectic
capacities. These capacities are invariant under taking products, and so are
well-suited for studying the stabilized embedding problem. The aim of this note
is to apply this theory, assuming its expected properties, to solve the
stabilized embedding problem for integral ellipsoids, when the eccentricity of
the domain is the opposite parity of the eccentricity of the target and the
target is not a ball. For the other parity, the embedding we construct is
definitely not always optimal; also, in the ball case, our methods recover
previous results of McDuff, and of the second named author and Kerman.
  There is a similar story, with no condition on the eccentricity of the
target, when the target is a polydisc: a special case of this implies a
conjecture of the first named author, Frenkel, and Schlenk concerning the
rescaled polydisc limit function. Some related aspects of the stabilized
embedding problem and some open questions are also discussed.
