Have you plotted your target density? You can use rstan to evaluate the log density at values of sigma on a grid and then plot them. Then you know what the sampler should be doing.
I think the bigger problem here is that you’re not defining a generative statistical model of the type sampling is intended to solve. Instead, you have some kind of constraint satisfaction problem, which may introduce an unintended distribution.
My general advice is to think about Bayesian models generatively. Here, you only have a single parameter sigma. With the parameter value and constants (here just the size N), you should be able to generate the modeled data of point_estimate, upper_bounds and cumulative_prob. Or maybe upper bounds are given as constants? It helps to clarify which variables are fixed and which are random.
To help you think about this, what is the model for different respondents returning different values for the location parameter? Presumably they’re generated with some noise around the true value, which I would suggest making latent. That is, introduce a location parameter mu and then assume that the respondents have something like point_estimate[n] ~ normal(mu, tau), where you can fit tau too. Then, given N, mu, tau, and sigma, you can generate the point_estimate. You’d also need an observation model for their cumulative probability estimates (those are also point estimates, which is why I would not recommend names like “point_estimate” for variables—here, I might use estimated_location if I wanted to be verbose or just y if I was being terse).
@andrewgelman often says that adding uncertainty to a deterministic model in this way “greases the wheels of commerce.” I think you’ll find it easier to reason about the model and you’ll find it easier to fit.