Nilpotency conjecture for generators of cyclotomic Khovanov–Lauda algebras at quantum characteristic zero

Let $R^{\Lambda}_{\alpha}$ be the cyclotomic Khovanov–Lauda algebra, with generators $y_1,\ldots,y_d$, and let ${\Lambda}$ be a dominant weight of level $l$. Assume that the quantum characteristic is $e=0$. Nilpotency conjecture. For every $1\leq r\leq d$, one has

$$y_r^l=0$$

in $R^{\Lambda}_{\alpha}$. The assertion is immediate from the defining relations when $l=1$ and was established for $l=2$ in the cited work; the authors state that they do not know how to verify even the level-two case using only the relations.