Canonical-basis characterization conjecture for totally nonnegative double flag varieties

Let $G$ be simply-laced, let $\lambda_1,\lambda_2\in X^{++}$ be regular dominant weights, and let $^{\omega}\Lambda_{\lambda_1}$ and $\Lambda_{\lambda_2}$ be the associated simple lowest- and highest-weight modules, with canonical basis $\mathbf B(^{\omega}\Lambda_{\lambda_1}\otimes\Lambda_{\lambda_2})$. Under the natural embedding

$$G/B^+\times G/B^-\longrightarrow \mathbb P(^{\omega}\Lambda_{\lambda_1}\otimes\Lambda_{\lambda_2}),$$

the set $\mathbb P(^{\omega}\Lambda_{\lambda_1}\otimes\Lambda_{\lambda_2})_{\geq0}$ consists of lines whose coordinates in this canonical basis are all nonnegative.

Canonical-basis characterization conjecture.

$$(G/B^+\times G/B^-)_{\geq0}=(G/B^+\times G/B^-)\cap\mathbb P(^{\omega}\Lambda_{\lambda_1}\otimes\Lambda_{\lambda_2})_{\geq0}.$$

This would give a representation-theoretic characterization of the totally nonnegative double flag variety analogous to the known characterization for totally nonnegative flag varieties. The supplied text presents the equality as the proposed description, but gives no resolution status.