  This paper fully answers a long standing open question concerning the
stability/instability of pure gravity periodic traveling water waves -- called
Stokes waves -- at the critical Whitham-Benjamin depth $
\mathtt{h}_{\scriptscriptstyle WB} = 1.363... $ and nearby values. We prove
that Stokes waves of small amplitude $ \mathcal{O}( \epsilon ) $ are, at the
critical depth $ \mathtt{h}_{\scriptscriptstyle WB} $, linearly unstable under
long wave perturbations. This is also true for slightly smaller values of the
depth $ \mathtt{h} > \mathtt{h}_{\scriptscriptstyle WB} - c \epsilon^2 $, $ c >
0 $, depending on the amplitude of the wave. This problem was not rigorously
solved in previous literature because the expansions degenerate at the critical
depth. In order to resolve this degenerate case, and describe in a
mathematically exhaustive way how the eigenvalues change their
stable-to-unstable nature along this shallow-to-deep water transient, we Taylor
expand the computations of arXiv:2204.00809v2 at a higher degree of accuracy,
derived by the fourth order expansion of the Stokes waves. We prove that also
in this transient regime a pair of unstable eigenvalues depict a closed figure
"8", of smaller size than for $ \mathtt{h} > \mathtt{h}_{\scriptscriptstyle WB}
$, as the Floquet exponent varies.
