Is it assumed that lower bounds on the size of monotone circuits apply to general Boolean circuits too?
Éva Tardos gave a function which can be computed by a polynomial size general circuit but requires an exponential size monotone circuit. The circuit computes a good enough approximation to the Lovász theta function of the input graph.

Razborov gave an $n^{\Omega(\log n)}$ lower bound monotone circuits computing the bipartite perfect matching function, for which polynomial size general circuits exist.