Unique ergodicity of minimal systems without nontrivial finite-independent pairs

Let $(X,T)$ be a minimal topological dynamical system, and let ${\rm Ind}_{fip}(X,T)$ denote its set of finite-independent pairs. Write $\Delta_X=\{(x,x):x\in X\}$. Unique ergodicity conjecture. If

$${\rm Ind}_{fip}(X,T)=\Delta_X,$$

then $(X,T)$ is uniquely ergodic. The paper has already shown that every ergodic invariant measure of such a system is measure-theoretically isomorphic to the Haar measure on an $\infty$-step nilsystem; unique ergodicity would therefore give the corresponding topological uniqueness of the invariant measure.