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Staged trees are a recently-developed, powerful family of probabilistic graphical models. An equivalence class of staged trees has now been characterised, and two fundamental statistical operators have been defined to traverse the equivalence class of a given staged tree. Here, two staged trees are said to be statistically equivalent when they represent the same set of distributions. Probabilistic graphical models such as staged trees are increasingly being used for causal analyses. Staged trees which are within the same equivalence class can encode very different causal hypotheses but data alone cannot help us distinguish between these. Therefore, in using score-based methods to learn the model structure and distributions from data for causal analyses, we should expect that a suitable scoring function is one which assigns the same score to statistically equivalent models. No scoring function has yet been proven to have this desirable property for staged trees. In this paper, we present a novel Bayesian Dirichlet scoring function based on path uniformity and mass conversation, and prove that this new scoring function is score-equivalent for staged trees.