The total positivity criterion from rigid modules

Let $X=B_K^-d4f G$ be the partial flag variety, let $X_{>0}$ be its totally positive part, and let $N_K$ be the embedded open cell. Let

$$T=T_1\oplus\cdots\oplus T_d$$

be a basic complete rigid $\Lambda$-module in $\operatorname{Sub}Q_J$, where $d=\dim X$. Total positivity conjecture. For $x\in N_K$, one has $x\in X_{>0}$ if and only if

$$\varphi_{T_i}(x)>0,\qquad (i=1,\ldots,d).$$

This proposes a cluster-theoretic description of total positivity using only $d$ inequalities; it is presented as an alternative to descriptions using sufficiently large dual canonical basis representations.