Algebraic monoids with affine group of invertible elements

Let $X$ be an algebraic variety with a monoid structure, i.e., there is an associative multiplication $X \times X \to X$, which is a morphism of algebraic varieties and admits a neutral element. Denote the group of invertible elements by $G(X)$. It is known that $G(X)$ is an algebraic group, open in $X$. It is easy to see that if $X$ is affine, then $G(X)$ is affine as well. We plan to prove the converse: if $G(X)$ is an affine algebraic group, then the variety $X$ is also affine. The talk is based on [1].
[1] Alvaro Rittatore. Algebraic monoids with affine unit group are affine. Transform. Groups 12 (2007), no. 3, 601-605