what do you hope to prove though? like Kyle said, gamemode and steam's gamemode (which is really gamescope) are not related at all.
lsof may not give you the whole picture either, at times processes will hold on to file handles of loaded dlls, at other times they will let go once the library has been memory mapped. Its opening a can of worms if you want to go that route for investigating whatever it is you hope to find out