Recent high-angular-resolution observations indicate the need for core growth to form high-mass stars. To understand the gas dynamics at the core scale in the very early evolutionary stages before being severely affected by feedback, we have conducted Atacama Large Millimeter/submillimeter Array (ALMA) observations toward a 70 mu m dark massive clump, G337.541-00.082 as part of the Global and Local infall in the ASHES sample (GLASHES) program. Using dense gas tracers such as N2H+ (J = 1-0) and HNC (J = 3-2), we find signs of infall from the position-velocity diagram and more directly from the blue asymmetry profile in addition to the clump-scale velocity gradient. We estimate infall velocities from intermediate and low-mass cores to be 0.28-1.45 km s-1, and infall rates to be on the order of 10-4-10-3 M circle dot yr-1, both are higher than those measured in low-mass star-forming regions by more than a factor of 5 and an order of magnitude, respectively. We find a strong correlation between the infall velocity with the nonthermal velocity dispersion, suggesting that infall may contribute significantly to the observed line width. Consistent with clump-fed scenarios, we show that the mass infall rate is larger for larger core masses and shorter distances to the clump center. Such high infall rates in cores embedded in IRDCs can be considered as strong signs of core growth, allowing high-mass star formation from intermediate-mass cores that would not initially form high-mass stars at their current mass.