The symmetry conjecture for the Solomon homomorphism

Let $W$ be a finite Coxeter group, and let $\Sigma W$ be the Solomon descent algebra of $W$. The Solomon homomorphism $[...]$ maps elements of $\Sigma W$ to class functions on $W$. Symmetry conjecture. For all $x,y\in \Sigma W$,

$$\Phi(x)(y)=\Phi(y)(x).$$

Jöllenbeck and Reutenauer proved this symmetry for Coxeter groups of type $A_n$; the conjecture asks for it for all finite Coxeter groups.