The leading-word conjecture for Lyndon loop root vectors

Let $\Delta^+$ be the set of positive roots, let $d\in\mathbb{Z}$, and let $\ell(\alpha,d)$ denote the Lyndon loop word of degree $(\alpha,d)$. Let $\Phi^L$ be the homomorphism from the quantum loop group to the corresponding shuffle algebra, and let a good loop word mean a loop word occurring in the relevant Lyndon-word construction; order loop words by the order used for leading words. Leading-word conjecture. For any $(\alpha,d)\in\Delta^+\times\mathbb{Z}$, the leading word of $\Phi^L(e_{\ell(\alpha,d)})$ is $\ell(\alpha,d)$. Moreover, the word $\ell(\alpha,d)$ is the smallest good loop word of degree $(\alpha,d)$. Computer experiments in all types, for a particular order of the simple roots, suggest this loop analogue of the corresponding finite-type lemma; the conjecture is intended to complete the triangularity argument underlying the PBW statement, but no proof or resolution is given here.