Before revision:  It turns out that the odds ratio is generally a sharp upper bound for counterfactual relative risk under some monotonicity assumptions, without resorting to strong ignorability, nor to the rare-disease assumption.
After revision:  It turns out that the odds ratio is generally a sharp upper bound for counterfactual relative risk under the monotone treatment response and monotone treatment selection assumptions, without resorting to strong ignorability, nor to the rare-disease assumption.