  Lacunar spinel GaTa$_4$Se$_8$ is a unique example of spin-orbit coupled Mott
insulator described by molecular $j_{\text{eff}}\!=\!3/2$ states. It becomes
superconducting at T$_c$=5.8K under pressure without doping. In this work, we
show, this pressure-induced superconductivity is a realization of a new type
topological phase characterized by spin-2 Cooper pairs. Starting from
first-principles density functional calculations and random phase
approximation, we construct the microscopic model and perform the detailed
analysis. Applying pressure is found to trigger the virtual interband tunneling
processes assisted by strong Hund coupling, thereby stabilizing a particular
$d$-wave quintet channel. Furthermore, we show that its Bogoliubov
quasiparticles and their surface states exhibit novel topological nature. To
verify our theory, we propose unique experimental signatures that can be
measured by Josephson junction transport and scanning tunneling microscope. Our
findings open up new directions searching for exotic superconductivity in
spin-orbit coupled materials.
