Quantum lifespan conjecture for the homogeneous complex Monge–Ampère equation

Let $S_T=[0,T]\times\mathbb{R}$ and let $\varphi_\infty$ be the quantum analytic continuation potential for the Cauchy data $(\omega_{\varphi_0},\dot{\varphi}_0)$. Define $T_{\operatorname{span}}$ as the supremum of the $T\geq 0$ for which the homogeneous complex Monge–Ampère Cauchy problem admits a $\pi_2^*\omega$-plurisubharmonic solution, and define $T_{\operatorname{span}}^Q$ as the supremum of the $T\geq 0$ for which $\varphi_\infty$ solves that problem. Quantum lifespan conjecture. The quantum analytic continuation potential $\varphi_\infty$ solves the homogeneous complex Monge–Ampère equation for as long as it admits a solution; equivalently,

$$T_{\operatorname{span}}^Q=T_{\operatorname{span}}.$$

This would provide a general method for solving the ill-posed Cauchy problem for the homogeneous complex Monge–Ampère equation for the full lifespan of a solution; the supplied text does not state whether the conjecture has been resolved.