Let $k$ be an algebraically closed field of characteristic zero and let $X$ be a toric variety over $k$, i.e. $X$ is a normal, irreducible $k$-variety and it admits an algebraic action of a torus with an open orbit.
Now comes the question: If $X$ is quasi-affine and smooth, is the $k$-algebra $k[X]$ of regular functions on $X$ always finitely generated (over $k$)? Is this true for arbitrary toric $X$?
I know, that in general (i.e. when $X$ is not necessarily toric), then $k[X]$ can be non-finitely generated, see e.g. the article of Winkelmann [Win03].
Any proof, counter-example or reference to my question would be very welcome!