You could use the lognormal distribution as a prior. Since it’s defined as the exponential of a normal random variable, its support is bounded to be positive. If you set its location parameter to 0, then it will have an expected value close to 1. I am not sure if there is a straightforward way to enforce the condition that up and down-weightings are equally probable; you could adjust the variance so that the probability on either side of 1 integrates to 1/2.
My personal opinion is that if the rescalings in either direction are equally likely in log-space (as is the case here), that’s good enough.