As mentioned by Igor Belegradek, the curvature gets destroyed by averaging.

Assume there is a modified averaging process that preserves positive curvature.
Then you the [Hopf conjecture][1] (there is no positively curved metric on $\mathbb{S}^2\times\mathbb{S}^2$) would follow from the Hsiang--Kleiner result. 
So at least it is too much to expect.


  [1]: https://en.wikipedia.org/wiki/Hopf_conjecture