Promotes the association of the proteasome activator complex subunit PSME3 with the 20S proteasome and regulates its activity. Inhibits PSME3-mediated degradation of some proteasome substrates, probably by affecting their diffusion rate into the catalytic chamber of the proteasome. Also inhibits the interaction of PSME3 with COIL, inhibits accumulation of PSME3 in Cajal bodies and positively regulates the number of Cajal bodies in the nucleus. . This is the function of PSME3IP1 (proteasome activator subunit 3 interacting protein 1, ENSG00000172775).