Regularity conjecture for complex Monge–Ampère equations with bounded or Hölder right-hand side

Let $\varphi$ solve the complex Monge–Ampère equation considered in the paper, with right-hand side $F$. The question concerns estimates for the regularity of $\varphi$ under assumptions on $F$. Regularity conjecture. Does $\varphi$ satisfy a $C^{1,\alpha}$ estimate when $F$ is only bounded, and a $C^{2,\alpha}$ estimate when $F\in C^{\alpha}$? The $C^{2,\alpha}$ estimate is known under additional regularity assumptions on the solution, but obtaining it without any extra regularity assumption remains open; the corresponding $C^{1,\alpha}$ estimate is presented as a natural expectation motivated by the real Monge–Ampère theory.