  Core-collapse explosions of massive stars leave behind neutron stars, with a
known diversity that includes the "Central Compact Objects" (CCOs). Typified by
the neutron star discovered near the centre of the Cas A supernova remnant
(SNR), CCOs have been observed to shine only in X-rays. To address their
supernova progenitors, we perform a systematic study of SNRs that contain a CCO
and display X-ray emission from their shock-heated ejecta. We make use of X-ray
data primarily using the Chandra X-ray observatory, complemented with
XMM-Newton. This study uses a systematic approach to the analysis of each SNR
aimed at addressing the supernova progenitor as well as the explosion
properties (energy and ambient density). After fitting for the ejecta
abundances estimated from a spatially resolved spectroscopic study, we compare
the data to six nucleosynthesis models making predictions on supernova ejecta
yields in core-collapse explosions. We find that the explosion models commonly
used by the astrophysics community do not match the ejecta yields for any of
the SNRs, suggesting additional physics, e.g. multi-dimensional explosion
models or updated progenitor structures, are required. Overall we find low-mass
($\leq$25 solar masses) progenitors among the massive stars population and
low-energy explosions ($<$10$^{51}$ ergs). We discuss degeneracies in our model
fitting, particularly how altering the explosion energy affects the estimate of
the progenitor mass. Our systematic study highlights the need for improving on
the theoretical models for nucleosynthesis predictions as well as for
sensitive, high-resolution spectroscopy observations to be acquired with
next-generation X-ray missions.
