The Chevalley basis Khovanskii basis conjecture

Let $\X=\G/\P$ be a homogeneous space with a dominant weight $\varpi$ yielding a projective embedding $\X\hookrightarrow\bP(V_\varpi)$, and let $\bs$ be a reduced expression for $\wP$. A Chevalley basis for $V_\varpi$ is the weight basis obtained from actions of monomials in Chevalley generators. The Chevalley basis Khovanskii basis conjecture. There exists a reduced expression $\bs$ such that the coordinates corresponding to any Chevalley basis for $V_\varpi$ form a Khovanskii basis for $\C[\X]$, and $\X$ admits a toric degeneration whose normalization is the toric variety associated to $\cP_{\X,\varpi,\bs}$. The conjecture extends the minuscule construction to arbitrary homogeneous spaces and embeddings; the authors note that the suitable choice of reduced expression and weight basis can affect whether the coordinate valuations generate the valuation semigroup.