Background dependence of local field operators inside a black hole and non renormalizability of gravity?

Question

In this physics blog article, it is explained that in quantum gravity there is no background independent description of local field operators inside a black hole

One can't background-independently localize field operators in QG

In particular the renormalization group analogy of how Polchinsky and  colleagues try to obtain a background independent definition of local  oparators inside a black hole picks me.  So could one also see that what  they try does not work from the fact, that gravity is not  renormalizable, such that for example in the same way as in a non  renormalizable theory the UV cutoff dependence of the IR limit of the  effective action can not be gotten rid of, the definition of local  operators inside a black hole have to depend on a choosen background?