  A large fraction of observed protoplanetary disks in nearby Star-Forming
Regions (SFRs) are fainter than expected in CO isotopologue emission. Disks not
detected in 13CO line emission are also faint and often unresolved in the
continuum emission at an angular resolution of around 0.2 arcseconds. Focusing
on the Lupus SFR, the aim of this work is to investigate whether the population
of CO-faint disks comprises radially extended and low mass disks - as commonly
assumed so far - or if it is of intrinsically radially compact disks, an
interpretation that we propose in this paper. The latter scenario was already
proposed for individual sources or small samples of disks, while this work
targets a large population of disks in a single SFR for which statistical
arguments can be made. A new grid of physical-chemical models of compact disks
has been run with DALI in order to cover a region of the parameter space that
had not been explored before with this code. Such models have been compared
with 12CO and 13CO ALMA observations of faint disks in Lupus. Disks that are
not detected in 13CO emission and with faint or undetected 12CO emission are
consistent with compact disk models. For radially compact disk, the emission of
CO isotopologues is mostly optically thick and it scales with the surface area:
i.e., it is fainter for smaller objects. The fraction of compact disks is
potentially between roughly 50% and 60% of the entire Lupus sample. Deeper
observations of 12CO and 13CO at a moderate angular resolution will allow us to
distinguish whether faint disks are intrinsically compact, or if they are
extended but faint, without the need of resolving them. If the fainter end of
the disk population observed by ALMA disk surveys is consistent with such
objects being very compact, this will either create a tension with viscous
spreading or require MHD winds or external processes to truncate the disks.
