Consider the setting of Theorem A, i.e. $\{\omega\}$ is the class of an (ample) line bundle $L$ on $X$. The strategy proposed in the preliminary work [KT00] to solve Monge-Ampère equations mostly deals with the case of a Dirac mass $\mu$ at a divisorial point $x\in X^{\mathrm{div}}$. The authors introduce the envelope (8.5), and assume by contradiction that $\MA(\varphi_{x})$ is not supported at $x$. They define a limit functional $F$ obtained by looking at the asymptotics of ball volumes in the space of sections of $mL$ as $m\to\infty$, and indicate that $F$ should satisfy $F(\varphi_{x}+\varepsilon f)=F(\varphi_{x})+\varepsilon\int f\MA(\varphi_{x})+O(\varepsilon^{2})$ for each $f\in C^{0}(X)$. Comparing with [BB10] in the complex case, $F$ is likely to coincide with $E_{\omega}\circ P_{\omega}$, so that a version of the differentiability property (Theorem 7.2) would also be a key ingredient in the approach proposed in [KT00].