I don't have the rep to post this as a comment, so I'll give it as an answer. I should say that I am not necessarily in this field; I am an astrophysicist.
I suspect that the idea of the Island of Stability is just a continuation of the early shell model for nuclei. This is basically a statement that nuclei with closed or filled energy shells are the most stable; this is analogous to the idea for atoms where the noble gases have filled electron orbitals, and don't interact as strongly as those atoms with unfilled orbitals.
The case of nuclei is more complicated because you have to consider the "orbitals" of two species: protons and neutrons. Furthermore, protons are charged while neutrons are neutral - which gives the two species slightly different behavior with regards to interpretation as shell structure. Nevertheless, there are so-called "Magic numbers" of protons and neutrons that give rise to the stable nuclei in the shell model. If you look at the valley of stability, you will see that the # protons = # neutrons, up to about an atomic number of 20, and then the valley turns over with stable nuclei having more neutrons than protons. There are several reasons for this turn over, some of which are related to the Coulomb repulsion between protons. This all likely is a deviation from the perfect shell structure.
If one continues the shell idea - perhaps taking into account the fact that protons are charged - past the known valley of stability, then there is a possibility of an island of stability, whereupon the next largest "stable shell" is filled. I think the fact that the simple shell model breaks down even at lower atomic number is an indication that it shouldn't' be used at the high atomic numbers implied in the island of stability.
I don't know of any group currently working on this region of the chart of nuclides. The closest thing I can think of is the upcoming Facility for Rare Isotope Beams (FRIB) project that will likely investigate very neutron rich material. The "Rare" in FRIB effectively means unstable, but they may be able to probe around in the region of the expected island of stability.