Artin groups' algebraic and $K(\pi,1)$ conjectures

Let $\Gamma$ be a finite simplicial graph, and let $A_{\Gamma}$ be the Artin group with generators the vertices of $\Gamma$ and braid relations determined by its edge coefficients. Call $A_{\Gamma}$ spherical when its associated Coxeter group is finite, and non-spherical otherwise.

Artin groups' algebraic and $K(\pi,1)$ conjectures. For every Artin group $A_{\Gamma}$: (1) $A_{\Gamma}$ has solvable word and conjugacy problems; (2) $A_{\Gamma}$ is torsion-free; (3) the centre of $A_{\Gamma}$ is trivial if $A_{\Gamma}$ is non-spherical and infinite cyclic if $A_{\Gamma}$ is spherical; and (4) $A_{\Gamma}$ satisfies the $K(\pi,1)$ conjecture.

These conjectures are known for spherical Artin groups, Artin groups of type FC, and Artin groups of dimension $2$, but remain open in general.