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

Assume there is a modified the averaging process that preserves positive curvature. Then you would prove Hopf conjecture (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.

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 (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.