Do we have a chance for renormalizable and unitary quantum gravity?

The main difficulty of perturbative quantum gravity (QG) in 4d is the conflict between renormalizability and unitarity. The simplest version of QG is based on General Relativity and is non-renormalizable. One can construct renormalizable and even superrenormalizable versions of QG by introducing higher derivatives, but then one has to deal with the unphysical higher derivative massive ghosts. The non-polynomial models of QG have no ghosts at the tree level, but taking loop corrections into account one meets infinite amount of ghost-like complex states. The same is true for the string-induced gravitational action, which requires an infinite amount of fine-tuning to become ghost-free. At the same time, the polynomial superrenormalizable versions of QG with complex poles have an attractive feature to be unitary within the Lee-Wick approach. These theories have unambiguous and exactly calculable beta-functions and even can be made finite. An interesting open question is whether these theories can be tested experimentally.