Toomre Q maps for the best disk candidates in the CORE survey assuming a protostar is located at the position of the continuum peak as depicted by a star and accounting for the self-gravity of the disk (protostellar and gas mass values are listed in Table 7). The contours correspond to the 1.37 mm continuum as described in Fig. 1. Regions outside of 6σ continuum contours are masked out. Fragments detected by ancillary observations are marked by plus symbols in green for AFGL2591 (Suri et al. 2021), IRAS 23385 (Cesaroni et al. 2019), and NGC 753 8IRS1 (Beuther et al. 2017). The synthesised beam is shown in the bottom left corner and a scale bar in the bottom right corner of each panel.