Yes, but a "range of prior distributions" doesn't mean every possible prior distribution. Sometimes, the information in the prior distribution is required to get you to a place where your computational system can efficiently explore a meaningful subspace instead of providing nonsense.

If meaningfully different priors lead to meaningfully different posteriors, you're probably missing something that would either eliminate one of those priors from contention or marry the differing behavior in some unifying explanation/model. Either way is a win in my book; both provide a new direction for research!