  We consider the Cauchy problem for the focusing nonlinear Schr\"odinger
equation with initial data approaching different plane waves
$A_j\mathrm{e}^{\mathrm{i}\phi_j}\mathrm{e}^{-2\mathrm{i}B_jx}$, $j=1,2$ as
$x\to\pm\infty$. The goal is to determine the long-time asymptotics of the
solution, according to the value of $\xi=x/t$. The general situation is
analyzed in [7] where we develop the Riemann-Hilbert approach and detect
different scenarios of asymptotic analysis, depending on the relationships
between the parameters $A_1$, $A_2$, $B_1$, and $B_2$. In particular, in the
shock case $B_1<B_2$, some scenarios include genus $3$ sectors, i.e., ranges of
values of $\xi$ where the leading term of the asymptotics is given in terms of
hyperelliptic functions attached to a Riemann surface $M(\xi)$ of genus three.
The present paper is devoted to the complete asymptotic analysis in such a
sector.
