Exponential monotone circuit complexity for an AC0 function

Determine whether there exists a monotone function in AC0 whose monotone circuit complexity is exponential.

Background

The paper proves superpolynomial monotone circuit lower bounds for a padded odd-factor function in AC0, specifically bounds of the form n^{\Omega(\log^k n)} for every fixed k. It explains that an exponential monotone circuit lower bound for an explicit AC0 function would yield an ultimate general-versus-monotone separation, extending the known qualitative separations discussed in the paper.