Constant-degree polynomial lower bound on a general polytope

Construct a constant-degree polynomial over some polytope for which the active-set method requires super-polynomially many iterations, irrespective of the pivot rule.

Background

The paper observes that increasing the complexity of the feasible region may permit a substantial reduction in the degree of the objective polynomial needed for an unconditional lower bound. The question asks whether a constant-degree objective, rather than a logarithmic- or linear-degree objective over the hypercube, can force super-polynomial behavior for every pivot rule on a suitably chosen polytope.