The Hessian rank conjecture for Coxeter arrangements of types E6, E7, and E8

Let $\mathsc{A}$ be the Coxeter arrangement associated with a finite reflection group $W$. The Hessian rank conjecture. The Hessian rank condition holds for $\mathsc{A}$ if $W$ is of type $E_6$, $E_7$, or $E_8$. Computing limitations leave this conjecture open for these exceptional types; the preceding theorem establishes the Hessian rank condition for all other types considered in the paper.