Lions' maximal regularity conjecture for viscous Hamilton-Jacobi equations

Consider the viscous Hamilton-Jacobi equation

$$--\Delta u+|Du|^\gamma=f(x)\quad\text{in }\Omega,$$

where $\Omega$ is a domain, $\gamma>1$, and $f\in L^q(\Omega)$ with $q>1$. Write $\gamma'=\frac{\gamma}{\gamma-1}$ for the conjugate exponent. Lions' conjecture. If

$$q>\frac{d}{\gamma'}=\frac{d(\gamma-1)}{\gamma},$$

then every solution satisfies the a priori estimate

$$\|D^2u\|_{L^q}+\||Du|^\gamma\|_{L^q}\leq C(\|f\|_{L^q},d,q,\gamma).$$

Moreover, the estimate is false when $q\leq d/\gamma'$. This conjecture concerns maximal $L^q$-regularity for viscous Hamilton-Jacobi equations with unbounded source terms; the source attributes it to P.-L. Lions. The supplied text does not indicate whether it has been resolved.