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https://link.springer.com/article/10.1007/s10479-010-0682-5 parallelizes https://www.researchgate.net/profile/Eduardo-Rodriguez-Tello/publication/258246465_Metaheuristics_for_the_Maximum_Parsimony_Problem/links/00463527d104eeddde000000/Metaheuristics-for-the-Maximum-Parsimony-Problem.pdf
Idea is to use NNI search, then when no NNI improvements are possible, to search SPR neighbourhood, then TBR neighbourhood. (We could add the SPR-but-only-on-leaves neighbourhood as an additional layer.) On finding a better tree in a new neighbourhood the process can recommence.
Built on by #170, which is closely linked.