  We explore both experimental and theoretical aspects of the superconducting
properties in the distinctive layered caged compound,
Ba$_{3}$Ir$_{4}$Ge$_{16}$. Our approach integrates muon spin rotation and
relaxation ($\mu$SR) measurements with magnetization and heat capacity
experiments, accompanied by first-principle calculations. The compound's bulk
superconductivity is unequivocally established through DC magnetization
measurements, revealing a critical temperature ($T_\mathrm{C}$) of 5.7 K. A
noteworthy characteristic observed in the low-temperature superfluid density is
its saturating behavior, aligning with the features typical of conventional
Bardeen-Cooper-Schrieffer (BCS) superconductors. The assessment of moderate
electron-phonon coupling superconductivity is conducted through transverse
field $\mu$SR measurements, yielding a superconducting gap to $T_\mathrm{C}$
ratio ($2\Delta(0)/k_\mathrm{B}T_\mathrm{C}$) of 4.04, a value corroborated by
heat capacity measurements. Crucially, zero field $\mu$SR measurements dismiss
the possibility of any spontaneous magnetic field emergence below
$T_\mathrm{C}$, highlighting the preservation of time-reversal symmetry. Our
experimental results are reinforced by first-principles density functional
calculations, underscoring the intricate interplay between crystal structure
and superconducting order parameter symmetry in polyhedral caged compounds.
This comprehensive investigation enhances our understanding of the nuanced
relationship between crystal structure and superconductivity in such unique
compounds.
