Tian's uniform estimate conjecture for degenerate complex Monge–Ampère equations

Let $(X,\omega_X)$ be a polarized compact connected Kähler manifold of complex dimension $n$, let $(Y,\omega_Y)$ be a compact irreducible Kähler space of complex dimension $m\leq n$, and let $\pi:X\rightarrow Y$ be a surjective holomorphic map. Let $0\leq f\in L\log^{n+\varepsilon}L(X,\omega_X^n)$ for some $\varepsilon>0$, with

$$1=\int_X f\omega_X^n.$$

For $t\in(0,1)$, set

$$K_t:=\{\pi^*\omega_Y+t\omega_X\}^n>0.$$

The solutions $\psi_t$ of the complex Monge–Ampère equations

$$(\pi^*\omega_Y+t\omega_X+i\partial\bar\partial\psi_t)^n=K_t f\omega_X^n$$

should satisfy Tian's uniform $L^{\infty}$-estimate. Namely,

$$\operatorname{Osc}(\psi_t):=\sup_X\psi_t-\inf_X\psi_t\leq C<+\infty$$

for all $t\in(0,1)$. This estimate is the conjecture of Tian referred to in the source; the surrounding text states that the authors' $L^{\infty}$-estimate completely solves it, while the parser supplies no independent resolution status.