Logarithmic-degree polynomial lower bound on the hypercube

Construct a polynomial of degree O(log n) over the n-dimensional hypercube for which the active-set method requires super-polynomially many iterations, irrespective of the pivot rule.

Background

The paper constructs degree-n polynomials over the n-dimensional hypercube that force the active-set method, starting at the origin, to visit all hypercube vertices and therefore take exponentially many iterations for every pivot rule. It also obtains super-polynomial lower bounds with degree ω(log n). The stated question asks whether the degree can be reduced to O(log n) while retaining a super-polynomial iteration bound.