Optimal exponent conjecture for fully nonlinear uniformly elliptic equations

Let $\lambda$ and $\Lambda$ be the ellipticity constants in Proposition 2.1, and let $\varepsilon$ denote the exponent in its $W^{2,\varepsilon}$ estimate. Optimal exponent conjecture. The optimal exponent in Proposition 2.1 is

$$\varepsilon=2(\Lambda/\lambda+1)^{-1}.$$

The preceding construction shows that the estimate fails when $(\Lambda/\lambda+1)\varepsilon>2$, so this exponent is an upper bound for the range established by that proposition. The source does not state whether the endpoint exponent is attainable, leaving the optimality claim open.