The identification of type $\widehat{A_1}$ generalized minors with block-Toeplitz minors

Let $\operatorname{SL}_2(\mathcal{L})$ be the loop group, let $T_g$ be the block-Toeplitz matrix associated with $g\in\operatorname{SL}_2(\mathcal{L})$, and let $\Delta^{(i)}_{m,n}$ denote the corresponding minors for parity $i\in\{0,1\}$ and non-negative integers $m,n$. The type $\widehat{A_1}$ generalized minors of Fomin–Zelevinsky are defined on this loop group.

Generalized-minor identification conjecture. The minors $\Delta^{(i)}_{m,n}$ are precisely the generalized minors of Fomin–Zelevinsky.

This would identify the block-Toeplitz minors introduced in the paper with the generalized minors used in the theory of total positivity and canonical bases for loop groups. The source does not provide evidence of a resolution, so the conjecture is recorded as open.