Specifying Toeplitz or Hankel as a set?

Is there a reason why one cannot specify Hankel or Toeplitz matrices as a set constraint - much like semidefinite? One can define a Hankel/Toeplitz variable and then set an equality constraint, but his seems like over-parametrising the problem.
Thanks, Roy

The reason is it was not implemented, and it almost certainly never will be, due to lack of development to add new features.