  Many years ago Baxter introduced an inhomogeneous two-dimensional classical
spin model, now called the $\tau_2(t)$ model with free boundary conditions, and
he specialized the resulting quantum spin-chain Hamiltonian in a special limit
to a simple clock Hamiltonian. Recently, Fendley showed that this clock
Hamiltonian can be expressed in terms of free "parafermions." Baxter followed
this up by showing that this construction generalizes to the more general
$\tau_2(t)$ model, provided some conjectures hold. In this paper, we will
compare the different notations and approaches enabling us to express the
Hamiltonians in terms of projection operators as introduced by Fendley. By
examining the properties of the raising operators, we are then able to prove
the last unproven conjecture in Baxter's paper left in our previous paper. Thus
the eigenvectors can all be written in terms of these raising operators.
