How often do you hear religious people talk about their religious beliefs as their "deepest beliefs"? Very often, I suspect - if you do a google search, you'll find that this is a common trope. Not only that, non-believers who are solicitous toward religion will often make a fuss about how we must not mock the "deepest beliefs" of religious people, or force them to act in ways that violate their "deepest beliefs". Again, you can do your own google search - I'm not going to drag out the evidence, but will merely remark that this is a very common trope.
I suppose some religious people do, indeed, hold their religious beliefs at a very deep level, but this can't be assumed. For some others, it might be mainly a matter of associating powerful or "deep" emotions with these beliefs. That's not quite the same thing.
I suspect that many religious believers do not hold beliefs about certain gods, or the authority of certain organisations and holy books, or the efficacy of certain practices for spiritual salvation or the like, or the truth of certain religious moral claims at the very deepest level.
Say I were able to demonstrate to a particular religious believer's satisfaction the truth of the proposition: "Either God does not exist or you have three arms." Imagine this is a person who has the usual human complement of only two arms, and is as aware of this as most of us, and in the same ways, etc.
I would then be in a position to continue with this tight little argument:
P1. Either God does not exist or you have three arms. (This has, ex hypothesi, been demonstrated.)
P2. You do not have three arms. (This seems like an incontrovertible empirical fact.)
C. God does not exist. (Disjunction elimination)
I suppose that some believers in this situation might conclude that they actually do have three arms. Some believers might abandon their acceptance of disjunction elimination (or vE or "or elimination") as a logical rule. Many, however, would believe in the seemingly incontrovertible empirical fact that they are two-armed, and in the intuitively compelling rule of disjunction elimination, at a more fundamental level than they believe in the existence of God.
Of course, such a person might still be very emotionally attached to the idea that God exists, and might look desperately at the previous demonstration of P1., looking to find a flaw in the demonstration. But what if P1. also turned out to be demonstrable using only a mix of intuitively compelling logical rules and seemingly incontrovertible empirical facts? I'm pretty sure that many believers would crack and admit that God doesn't exist after all. For them, belief in the existence of God is not be as deeply held a belief as belief in certain everyday and seemingly incontrovertible empirical facts. Nor is it as deeply held as their belief in certain fundamental logical rules (whether or not they could actually name those rules formally).
I actually doubt that belief in God - or any other religious claims - is as deeply held for many people as belief in various everyday, seemingly incontrovertible facts, or as deeply held as belief in basic logical rules.
Unfortunately, no one has ever been able to demonstrate a proposition quite so devastating to theism as "Either God does not exist or you have three arms." The problem, then, is not necessarily that the religious beliefs of religious believers are literally their most deeply held beliefs. It's more that it's difficult devising an argument which shows a clear contradiction between religious beliefs and what are probably (even?) more deeply held beliefs.
All that said, it is worth watching how religious believers respond when confronted with, say, the Problem of Evil.
Some people don't want to look too closely at it, perhaps sensing how devastating it is to certainreligious and propositions that they hold dear. Some people are terribly troubled by it (and some of these actually do abandon their religious beliefs). Conversely, some people actually do dig in, and they often end up having to believe stuff that looks crazy to most non-believers... and would probably look crazy to many believers as well. This last class of people, those who dig in, perhaps do have their religious beliefs among those that are most deeply held. But that can end up looking a bit nuts, or seeming rather unpleasant, or as if they are occupying very contrived and unattractive positions.
And these religious people may not be typical, which is one reason why many people might simply not want to look closely.
And here we have a clue why some people actually do abandon religious beliefs. For them, these were not their deepest beliefs after all.