Principal ideal conjecture for greedy factorizations

Let $\Omega$ be the space of totally positive loop-group elements. For $X\in\Omega$, let $J(X)$ be the set of equivalence classes $[\mathbf i]$ of infinite reduced words $\mathbf i$ such that $X$ has a greedy factorization $e_{\mathbf i}(\mathbf a)$. Greedy principal ideal conjecture. The ideal $J(X)$ is principal. The source notes that $J(X)$ is an ideal in limit weak order, but does not establish principality; it also asks whether $J(X)=I(X)$.