Looking for a possible multi-column index
No wonder you get stuck. You are facing contradictory requirements, which a b-tree index cannot easily reconcile. You can hardly optimize the search for date and score at the same time. If you order by date, the scores are spread out arbitrarily - or vice versa.

There is a way though. You can make use of a set of partial indexes.
Consider the high-end solutions under this closely related question.