Universality conjecture for the word $u_N$ in the 0-Hecke algebra

Let $H_0(S_N)$ be the 0-Hecke algebra of the symmetric group, and let

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

For a signed diagram $D$ on the letters $1,\ldots,N-1$, let $u_N^D$ be the masked word obtained by applying the sign of each letter in $D$ to its occurrences in $u_N$. A word is universal if it contains every letter in $\{1,\ldots,N-1\}$ at least once and its masked word is demipotent for every signed diagram.

Universality conjecture. The word $u_N$ is universal.

This conjecture proposes a uniformly describable family of demipotents whose masked versions remain demipotent for every choice of signs. It is based on computer exploration, and the source does not provide a proof or resolution.