The CPT symmetry is one of the most fundamental symmetries in physics. Any violation of this symmetry would have profound implications for our understanding of the universe. In this study, we report the CPT symmetry tests in 3$\gamma$ decays of polarised $^3$S$_{1}$ positronium using the Jagiellonian Positron Emission Tomography device. The J-PET experiment allows sensitive and precise tests of CPT symmetry by measuring the angular correlation between the spin of ortho-positronium and the momenta direction of the annihilation photons emitted in ortho-positronium decay. The potential of J-PET in determining the full range of the expectation value of this correlation has improved the precision of the CPT symmetry test to 10$^{-4}$ already. The accuracy of this previous measurement was limited by statistics only. The new test is based on the increased statistics due to the modified experimental setup aiming at the improvement of detection efficiency and due to the usage of different positronium production chambers. The high precision of this test would open the possibility of exploring the limits of CPT symmetry validity in the charged leptonic sector. In this work, we discuss the evaluation of background sources in the CPT symmetry test with J-PET which is a crucial factor in improving the precision of such studies.