We explore the evolution of the structural phase transition of $rm{(Sr, Ca)_3Ir_4Sn_{13}}$, a model system to study the interplay between structural quantum criticality and superconductivity, by means of high-energy x-ray diffraction measurements at high pressures and low temperatures. Our results confirm a rapid suppression of the superlattice transition temperature $T^*$ against pressure, which extrapolates to zero at a critical pressure of $approx 1.79(4)$ GPa. The temperature evolution of the superlattice Bragg peak in $rm{Ca_3Ir_4Sn_{13}}$ reveals a drastic decrease of the intensity and an increase of the linewidth when $T rightarrow 0$ K and $p rightarrow p_c$. Such anomaly is likely associated to the emergence of quantum fluctuations that disrupt the formation of long-range superlattice modulation. The revisited temperature-pressure phase diagram of $rm{(Sr, Ca)_3Ir_4Sn_{13}}$ thus highlights the intertwined nature of the distinct order parameters present in this system and demonstrates some similarities between this family and the unconventional superconductors.
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