Sign-control conjecture for dual canonical basis immanants

Let $\operatorname{Imm}_v X$ be a Kazhdan–Lusztig immanant that is $k$-positive, and let $R,C$ be index sets for which the dual canonical basis element $\operatorname{Imm}_v X(R,C)$ is defined. Sign-control conjecture. If $\operatorname{Imm}_v X(R,C)$ is not identically zero, then $\operatorname{Imm}_v X(R,C)$ is $k$-positive.

The conjecture formalizes the observed agreement between the positivity of ordinary Kazhdan–Lusztig immanants and their dual canonical basis analogues; its status is not resolved in the supplied text.