The principal ideal conjecture for factorizations of totally positive loop-group elements

Let $\Omega$ be the space of totally positive loop-group elements. For $X\in\Omega$, let $I(X)$ be the set of equivalence classes $[\mathbf i]$ of infinite reduced words $\mathbf i$ such that $X\in E_{\mathbf i}$; this is a lower order ideal in the limit weak order. Principal ideal conjecture. For any $X\in\Omega$, the ideal $I(X)$ is a principal ideal. This conjecture would organize all infinite reduced-word factorizations of $X$ under the limit weak order; its status is not resolved in the source.