According to a recent paper by Peter Raven and collaborators, the Neotropics is the most biodiverse tropical region in absolute terms, encompassing one third of global diversity, but Southeast Asia is proportionally richer if we consider its smaller area.
Reference: Raven PH, Gereau RE, Phillipson PB, Chatelain C, Jenkins CN, Ulloa C. 2020. The distribution of biodiversity richness in the tropics. Science Advances 6, eabc6228.
Abstract: We compare the numbers of vascular plant species in the three major tropical areas. The Afrotropical Region (Africa south of the Sahara Desert plus Madagascar), roughly equal in size to the Latin American Region (Mexico southward), has only 56,451 recorded species (about 170 being added annually), as compared with 118,308 recorded species (about 750 being added annually) in Latin America. Southeast Asia, only a quarter the size of the other two tropical areas, has approximately 50,000 recorded species, with an average of 364 being added annually. Thus, Tropical Asia is likely to be proportionately richest in plant diversity, and for biodiversity in general, for its size. In the animal groups we reviewed, the patterns of species diversity were mostly similar except for mammals and butterflies. Judged from these relationships, Latin America may be home to at least a third of global biodiversity.