We present a polynomial-time algorithm for finding an optimal set of taxa that maximizes the weighted sum of the phylogenetic diversity across two phylogenetic trees. This resolves one of the challenges proposed as part of the Phylogenetics Programme held at the Isaac Newton Institute for Mathematical Sciences (Cambridge, 2007). It also completely closes the gap between optimizing phylogenetic diversity on one tree, which is known to be in P, and optimizing phylogenetic diversity across three or more trees, which is known to be NP-hard.