Cluster algebra structure on the homogeneous coordinate ring of a partial flag variety

Let $G$ be a semisimple complex algebraic group, let $P_K^-$ be the corresponding opposite parabolic subgroup, and write ${\mathbb C}[G/P_K^-]$ for the homogeneous coordinate ring. Let $\widehat{\mathcal{A}}$ be the lifted cluster algebra, with initial seed

$$\left(\left\{\widetilde{D}_{\varpi_{i_k},w_{\leq k}\varpi_{i_k}}\right\}\sqcup\left\{\Delta_{\varpi_j,\varpi_j}\mid j\in J\right\},\widehat{B}\right).$$

Cluster-structure conjecture. The homogeneous coordinate ring equals the lifted cluster algebra:

$${\mathbb C}[G/P_K^-]=\widehat{\mathcal{A}}.$$

In particular, ${\mathbb C}[G/P_K^-]$ is a cluster algebra with the displayed initial seed. The parser supplies no resolution status; the claim appears as a conjecture environment, although the surrounding text presents it as a result, so its status should be checked against the full paper.