  Nodal topological superconductors characterized by $p_x$-wave pairing
symmetry host flat-band Majorana bound states causing drastic anomalies in
low-energy electromagnetic responses. Nevertheless, the study of flat-band
Majorana bound states has been at a standstill owing to a serious lack of
candidate materials for $p_x$-wave superconductors. In this paper, by expanding
a scheme of planar topological Josephson junctions, we propose a promising
device realizing an effective $p_x$-wave superconductor. Specifically, we
consider a three-dimensional Josephson junction consisting of a thin-film
semiconductor hosting a persistent spin-helix state and two conventional
$s$-wave superconductors. We analytically obtain a topological phase diagram
and numerically demonstrate the emergence of flat-band Majorana bound states by
calculating the local density of states.
