  Spin-orbit coupling (SOC) often gives rise to interesting electronic and
magnetic phases in an otherwise ordinary pool of paramagnetic heavy metal
oxides. In presence of strong SOC, assumed to be working in $j$-$j$ coupling
regime, 5$d^4$ iridates are generally speculated to possess a nonmagnetic
$J_{eff}$~=~0 singlet ground state, which invariably gets masked due to
different solid-state effects (e.g. hopping). Here, we try to probe the
trueness of the atomic SOC-based proposal in an apparently 1-dimensional
system, Sr$_3$NaIrO$_6$, possessing a 2$H$ hexagonal structure with well
separated Ir$^{5+}$ (5$d^4$) ions. But all the detailed experimental as well as
theoretical characterizations reveal that the ground state of Sr$_3$NaIrO$_6$
is not nonmagnetic, rather accommodating a significantly high effective
magnetic moment on Ir$^{5+}$ ion. However our combined dc susceptibility
($\chi$), ${}^{23}$Na nuclear magnetic resonance (NMR),
muon-spin-relaxation/rotation ($\mu$SR) and heat capacity ($C_p$) measurements
clearly refute any sign of spin-freezing or ordered magnetism among the
Ir$^{5+}$ moments due to geometrical exchange frustration, while in-depth
zero-field (ZF) and longitudinal field (LF) $\mu$SR investigations strongly
point towards inhomogeneous quantum spin-orbital liquid (QSOL)-like ground
state. In addition, the linear temperature dependence of both the NMR
spin-lattice relaxation rate and the magnetic heat capacity at low temperatures
suggest low-lying gapless spin excitations in the QSOL phase of this material.
Finally, we conclude that the effective SOC realised in $d^4$ iridates are
unlikely to offer a ground state which will be consistent with a purely atomic
$j$-$j$ coupling description.
