The universal-word conjecture for demipotents in the 0-Hecke algebra

Let $I$ be the index set, let $D$ be a signed diagram, and let $w^D$ be the masked word obtained by applying the sign of each index in $D$ to every occurrence of that index in $w$. A word is universal if it contains every letter in $I$ at least once and $w^D$ is demipotent for every signed diagram $D$. For $N$, define

$$u_N=(1,2,\ldots,N-2,N-1,N-2,\ldots,2,1).$$

Universal-word conjecture. The word $u_N=(1,2,\ldots,N-2,N-1,N-2,\ldots,2,1)$ is universal. This conjecture proposes a uniform, simple expression whose signed versions produce demipotents in $H_0(S_N)$. The supplied text gives computer exploration as motivation but does not state a resolution.