The star formation rate in the Central Molecular Zone (CMZ) is an order of magnitude lower than predicted according to star formation relations that have been calibrated in the disc of our own and nearby galaxies. Understanding how and why star formation appears to be different in this region is crucial if we are to understand the environmental dependence of the star formation process. Here, we present the detection of a sample of high-mass cores in the CMZs dust ridge that have been discovered with the Submillimeter Array as part of the CMZoom survey. These cores range in mass from ~ 50 - 2150 Msun within radii of 0.1 - 0.25 pc. All appear to be young (pre-UCHII), meaning that they are prime candidates for representing the initial conditions of high-mass stars and sub-clusters. We report that at least two of these cores (c1 and e1) contain young, high-mass protostars. We compare all of the detected cores with high-mass cores in the Galactic disc and find that they are broadly similar in terms of their masses and sizes, despite being subjected to external pressures that are several orders of magnitude greater - ~ 10^8 K/cm^3, as opposed to ~ 10^5 K/cm^3. The fact that > 80% of these cores do not show any signs of star-forming activity in such a high-pressure environment leads us to conclude that this is further evidence for an increased critical density threshold for star formation in the CMZ due to turbulence.