I thought about it for so long that I started getting confused.
Final conclusion, I added his/her/their because I think in Ukrainian people tend to drop the pronoun when the context is clear.
E.g. I would totally say "Я шукаю валізу" rather than "Я шукаю свою валізу" cause I'm lazy; I'm not very sure if it works in third person too (Він шукає валізу instead of Він шукає свою валізу) just because I thought of it too much and lost the feeling xD
Added it! Thanks a lot for commenting!