Here is a hypothesis: A mature neuro-science will, as the eliminativists claim, have no room for concepts like "belief", "desire", etc. Suppose this hypothesis proves true. What should we then do? Obviously, it would be absurd to deny that we have beliefs or desires. Instead, we should deny that belief, desire, etc. occur in the neural system, which is what the neuro-science studies, and hold that they occur elsewhere. Since there is no other plausible candidate for the mind in the physical world besides the neural system, we should conclude that belief, desire, etc. occur outside the physical world, i.e., that some form of dualism is true. Moreover, I suspect that a neuro-science that would have no room for beliefs and desires would also have no room for the idea that there are states of the brain on which beliefs and desires supervene. Thus, it would lead us to a non-supervenient dualism.
But this is just an exercise in hypotheticality, since we are in no position to make such specific predictions about future science.