ap.analysis of pdes – Regularity of Aleksandrov solution to Monge-Ampère equation with infinite slope at boundary

I’m interested in the regularity of solutions to Monge-Ampère equations in a bounded convex domain $Omegasubsetmathbb{R}^n$. It seems that the following statement can be deduced from known results:

Proposition. Let $uin C^0(overline{Omega})$ be a convex function satisfying the following conditions:

(a) the Monge-Ampère measure of $u$ in $Omega$ is a positive smooth function times the Lebesgue measure;

(b) $u$ has infinite slope at every boundary point (or equivalently, the graph of $u$ does not have non-vertical supporting planes at any boundary point).

Then $u$ is smooth in $Omega$.

However, since my understanding of the known results is rather limited and superficial, I’m not sure whether this is correct. Here’s my argument: First, by Evans-Krylov theory, if $u$ is strictly convex , then condition (a) would imply the smoothness (see e.g. Theorem 3.1 in this survey of Trudinger-Wang). If $u$ is not strictly convex, then there is a supporting affine function $a$ with $ugeq a$ such that the closed convex set ${u=a}$ is not a single point. But it follows from “balancing of sections” that this set cannot have extremal points in $Omega$, hence must meet $partialOmega$ (see e.g. Theorem 7 in these notes of Connor Mooney, which violates condition (b). Q.E.D.

So my question is: Are the above proposition and argument correct? If yes, what are the references to cite if I want to quote the proposition in a paper?