4 problems

Neighborhood reduction conjecture. Suppose the relations $\hat{\mathscr{B}}\cup\mathscr{C}$ span $\hat{\mathcal{R}}_*(w)$ for each neighborhood $J\subset S$ with $J=J^*$ and each…

Finite generalized half-braid spanning-set conjecture. There exists a finite subset $\mathscr{C}$ of the generalized half-braid relations for $(W,S,*)$ such that the relations…

Perfectly braided twisted Coxeter system conjecture. A twisted Coxeter system $(W,S,*)$ is perfectly braided if $s^*\neq s$ for all $s\in S$.

Artin relation periodicity claim.