The theorem I am stating is Theorem 6.2 (Blessing of Binary) of the dissertation THE ROLE OF EXTREME POINTS FOR CONVEX HULL OPERATIONS, which was cited from Stochastic Dual Dynamic Integer Programming albeit with a different formalism.
Let $c\in \mathbb R^n,\tilde c\in \mathbb R^p, q\in \mathbb R^l$, and $A\in \mathbb R^{m\times n}, B \in \mathbb R^{m\times p}, C\in \mathbb R^{m\times l}$.
Let $\tilde X:=\{(x,y,z)\in \mathbb R^n\times [0,1]^p\times \{0,1\}^l\mid Ax+By+Cz\le b\}$.
For $\lambda\in[0,1]^p$, we define the optimization problem
$$ \phi(\lambda) := \min_{ \begin{aligned}(x,\lambda,z)&\in \tilde X\end{aligned} } c^T x + \tilde c^T \lambda+q^T z $$
So, if $z$ wasn't a binary vector, $\phi(\lambda)$ would be a simple linear optimization problem over the two vectors $x$ and $z$.
Next, we define the convex regularization ${\phi^*}^*$, in which we replace $\tilde X$ by its convex hull $\text{conv}(\tilde X)$: $$ \phi^{**}(\lambda) := \min_{ \begin{aligned}(x,\lambda,z)&\in {\text{conv}}(\tilde X)\end{aligned} } c^T x + \tilde c^T \lambda+q^T z $$ So, the feasible set of $\phi^{**}(\lambda)$ is the cut $\text{conv}(\tilde X)\cap \{(x,y,z)\in\mathbb R^n\times \mathbb R^p\times \mathbb R^l\mid y=\lambda\}$, both of which are convex, so $\phi^{**}$ is a convex optimization problem.
Now, the blessing of binary says, that for all binary vectors $\hat \lambda\in \{0,1\}^l$, we have that $$ \phi^{**}(\hat \lambda) =\phi(\hat \lambda) $$
However, since we can set $\hat\lambda =0$, doesn't this effectively say that we can reduce the MILP $$ \min_{ \begin{aligned}x\in \mathbb R^n, z\in \{0,1\}^l\\ Ax+Cz\le b\end{aligned} } c^T x +q^T z $$ , which looks like the most general MILP to me since $A$ can both be freely chosen, to a convex problem?