  We continue the investigation of the existence of absolutely continuous
(a.c.) spectrum for the discrete Schr\"odinger operator $\Delta+V$ on
$\ell^2(\Z^d)$, in dimensions $d\geq 2$, for potentials $V$ satisfying the long
range condition $n_i(V-\tau_i ^{\kappa}V)(n) = O(\ln^{-q}(|n|))$ for some
$q>2$, $\kappa \in \N$, and all $1 \leq i \leq d$, as $|n| \to \infty$. $\tau_i
^{\kappa} V$ is the potential shifted by $\kappa$ units on the $i^{\text{th}}$
coordinate. The difference between this article and \cite{GM2} is that here
finite linear combinations of conjugate operators are constructed leading to
more bands of a.c.\ spectrum being observed. The methodology is backed
primarily by graphical evidence because the linear combinations are built by
numerically implementing a polynomial interpolation. On the other hand an
infinitely countable set of thresholds, whose exact definition is given later,
is rigorously identified. Our overall conjecture, at least in dimension 2, is
that the spectrum of $\Delta+V$ is void of singular continuous spectrum, and
consecutive thresholds are endpoints of a band of a.c. spectrum.
