Truly exponential monotone circuit lower bounds for the clique function

Prove a truly exponential monotone circuit lower bound of the form 2^{\Omega(n)} for the clique function.

Background

The paper identifies the clique function as the most extensively studied explicit function for monotone circuit lower bounds. At the time of writing, the largest explicit lower bound is 2^{n^{1/2-o(1)}}, while truly exponential lower bounds are known for monotone formulas. Establishing a 2^{\Omega(n)} lower bound for monotone circuits would therefore substantially advance the search for explicit functions with maximal monotone circuit complexity.