Hilbert's fourteenth problem for the zip-form algebra

Let $k$ be the ground field and let $R_{\rm zip}$ be the $k$-algebra of functions associated with the zip datum. Hilbert finite-generation conjecture. The $k$-algebra $R_{\rm zip}$ is finitely generated. This is presented in the source in the context of Hilbert's finite-generation conjecture. The statement is refuted in general by Nagata's counterexample to Hilbert's original conjecture.