This post is about Natural Proofs barrier in computational complexity.

There are two recent papers related to this. They are:

Amplifying lower bounds by means of self-reducibility by Eric Allender and Michal Koucký,

Almost-natural proofs by Timothy Chow.

Alexander Razborov mentioned them at the end of his talk in the first Barriers Workshop at IAS in 2009. The video of his talk is available here. He mentioned that a possible way to respond to these would be to make the largeness condition relative. So here is my question:

Has there been any progress on this?