Not when they are serviced correctly...
I just replaced 2 sets of these on watches last week, and they should not press in as easy as that one did (with your fingers) as they are an interference fit that requires a press to install. Also as Jim noted, Omega specs call for Loctite retaining compound to be applied, so it's possible that was not done, or things were not properly cleaned/prepared before application.
Looking at the exact same model I just replaced the pushers on, it appears that the lower pusher may not be fully seated, as I can see what looks like a gap there between the shoulder on the pusher tube and the case - a clear shot of both pushers from the back side of the watch will show the situation better if you can post one. The angle makes it difficult to tell, but it seems to be sitting higher than normal from the case (bottom of the pusher cap should be slightly below the cut out in the case, and it looks even with the case in that angle).
As noted the watch is not water resistant now, and there is a chance debris (either external or residue from the Loctite if it was applied) has entered the case/movement. Omega would likely insist on a service if you brought it to them.
Cheers, Al
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