Variable-right-hand-side sigma-2 over sigma-1 estimates in dimension four

Determine whether purely local \(C^2\) estimates hold for strictly 2-convex solutions of the Hessian quotient equation \(\sigma_2(D^2u)/\sigma_1(D^2u)=\psi(x,u)\) in dimension \(n=4\) when the right-hand side is nonconstant.

Background

The paper explains that interior estimates for the $\sigma_2/\sigma_1$ Hessian quotient are known in dimension three for variable right-hand sides and in several settings through transformations or doubling arguments.

It specifically identifies the four-dimensional strictly 2-convex case with a nonconstant right-hand side as unresolved. The issue is whether the known methods can provide the same local estimates in this remaining case.