  In recent papers we have refined a conjecture of Lehrer and Solomon
expressing the character of a finite Coxeter group $W$ acting on the $p$th
graded component of its Orlik-Solomon algebra as a sum of characters induced
from linear characters of centralizers of elements of $W$. Our refined
conjecture relates the character above to a component of a decomposition of the
regular character of $W$ related to Solomon's descent algebra of $W$. The
refined conjecture has been proved for symmetric and dihedral groups, as well
as finite Coxeter groups of rank three and four.
  In this paper, the second in a series of three dealing with groups of rank up
to eight (and in particular, all exceptional Coxeter groups), we prove the
conjecture for finite Coxeter groups of rank five and six, further developing
the algorithmic tools described in the previous article. The techniques
developed and implemented in this paper provide previously unknown
decompositions of the regular and Orlik-Solomon characters of the groups
considered.
