Cluster-variable conjecture for dual canonical basis immanants

Fix $m$, and let $G^{w_0,w_0}$ denote the big open double Bruhat cell in $SL_m$. For $v\in S_n$, let $\operatorname{Imm}_v X(R,C)$ denote the associated dual canonical basis immanant, with $R,C\in\binom{[m]}{n}$ and the indicated admissibility condition when required. Cluster-variable conjecture. (1) Every cluster variable of $\mathbb C[G^{w_0,w_0}]$ is of the form $\operatorname{Imm}_v X(R,C)$ for some $v$ avoiding $2143$ and $1324$. (2) If $v$ avoids $2143$ and $1342$, $R,C$ are such that $\Gamma[v,w_0]$ is $(R,C)$-admissible, then $\operatorname{Imm}_v X(R,C)$ is a cluster variable when it is irreducible, and is a cluster monomial otherwise.

The conjecture concerns the relationship between dual canonical basis elements and the cluster algebra structure of the open double Bruhat cell. The supplied text gives no resolution status.