1) Symmetry Region 6 uses the same face as Region 4, but points in the wrong direction. Delete it. Except after I did step 2, I redefined all the Symmetry regions.
2) Open the Geometry in SpaceClaim. Use the Repair tab and simplify the surfaces, remove extra edges and extra points. On the Workbench tab, use the Share button. This will allow the mesh to connect across the different bodies without using Bonded Contact.
3) I added a Frictional Contact to the opening that is passing through.
4) There is no need for 20 steps. You can use 1 step to turn on one pressure and step 2 to turn on another pressure.
You should edit your geometry to make the base of the small cubic cavity have a thinner bottom wall thickness so it takes less pressure to close the valve.
I don't claim that the archive below will converge to the end because I can't run this on my laptop which has only 16 GB RAM. I might try to run it on my desktop with 64 GB of RAM later.