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The RADiCAL Collaboration is conducting R\&D on high performance electromagnetic (EM) calorimetry to address the challenges expected in future collider experiments under conditions of high luminosity and/or high irradiation (FCC-ee, FCC-hh and fixed target and forward physics environments). Under development is a sampling calorimeter approach, known as RADiCAL modules, based on scintillation and wavelength-shifting (WLS) technologies and photosensor, including SiPM and SiPM-like technology. The modules discussed herein consist of alternating layers of very dense (W) absorber and scintillating crystal (LYSO:Ce) plates, assembled to a depth of 25 $X_0$. The scintillation signals produced by the EM showers in the region of EM shower maximum (shower max) are transmitted to SiPM located at the upstream and downstream ends of the modules via quartz capillaries which penetrate the full length of the module. The capillaries contain DSB1 organic plastic WLS filaments positioned within the region of shower max, where the shower energy deposition is greatest, and fused with quartz rod elsewhere. The wavelength shifted light from this spatially-localized shower max region is then propagated to the photosensors. This paper presents the results of an initial measurement of the time resolution of a RADiCAL module over the energy range 25 GeV $\leq$ E $\leq$ 150 GeV using the H2 electron beam at CERN. The data indicate an energy dependence of the time resolution that follows the functional form: $\sigma_{t} = a/\sqrt{E} \oplus b$, where a = 256 $\sqrt{GeV}$~ps and b = 17.5 ps. The time resolution measured at the highest electron beam energy for which data was currently recorded (150 GeV) was found to be $\sigma_{t}$ = 27 ps.