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We use data from the pilot observations of the EMU/POSSUM surveys to study the 'missing supernova remnant (SNR) problem', the discrepancy between the number of Galactic SNRs that have been observed, and the number that are estimated to exist. The Evolutionary Map of the Universe (EMU) and the Polarization Sky Survey of the Universe's Magnetism (POSSUM) are radio sky surveys that are conducted using the Australian Square Kilometre Array Pathfinder (ASKAP). We report on the properties of seven known SNRs in the joint Galactic pilot field, with an approximate longitude and latitude of 323° ≤ l ≤ 330° and −4° ≤ b ≤ 2°, respectively, and identify 21 SNR candidates. Of these, four have been previously identified as SNR candidates, three were previously listed as a single SNR, 13 have not been previously studied, and one has been studied in the infrared. These are the first discoveries of Galactic SNR candidates with EMU/POSSUM and, if confirmed, they will increase the SNR density in this field by a factor of 4. By comparing our SNR candidates to the known Galactic SNR population, we demonstrate that many of these sources were likely missed in previous surveys due to their small angular size and/or low surface brightness. We suspect that there are SNRs in this field that remain undetected due to limitations set by the local background and confusion with other radio sources. The results of this paper demonstrate the potential of the full EMU/POSSUM surveys to uncover more of the missing Galactic SNR population. [ABSTRACT FROM AUTHOR]