Finkelberg–Fourier–Littelmann finite-generation conjecture for essential monomials

Let $S$ be a birational sequence and let $>$ be a monomial ordering. The construction associates to a dominant weight $\lambda$ a monoid $\Gamma(S,>,\lambda)$ of essential monomials, and the global monoid is

$$\Gamma(S,>):= \sum_{\lambda} \Gamma(S,>,\lambda).$$

Finkelberg–Fourier–Littelmann conjecture. For every birational sequence $S$ and monomial ordering $>$, the monoid $\Gamma(S,>,\lambda)$ is finitely generated.

The conjecture concerns the existence of finite generating sets for the monoids parametrizing monomial bases. It is stated as the main conjecture of the paper; the supplied text gives no evidence that it has been resolved.