Optimality of the fourth-power stability exponent

Determine whether the fourth-power dependence on the \(W^{1,4}\) distance in the global quantitative \(\sigma_2\)-stability estimate for normalized conformal metrics on a fixed nonround positive Einstein manifold is optimal.

Background

The global stability theorem proves that the $\sigma_2$-energy deficit controls the squared $H^1$ distance and the fourth power of the $W^{1,4}$ distance from the normalized Einstein background. The remark establishes that the quadratic $H^1$ exponent cannot be lowered, but it leaves unresolved whether the fourth-power exponent in the $W^{1,4}$ term can be improved or is sharp.