  The Skyrme model is extended with a sextic derivative term - called the
BPS-Skyrme term - and a repulsive potential term - called the loosely bound
potential. A large part of the model's parameter space is studied for the
4-Skyrmion which corresponds to the Helium-4 nucleus and emphasis is put on
preserving as much of the platonic symmetries as possible whilst reducing the
binding energies. We reach classical binding energies for Helium-4 as low as
0.2%, while retaining the cubic symmetry of the 4-Skyrmion, and after taking
into account the quantum mass correction to the nucleon due to spin/isospin
quantization, we get total binding energies as low as 3.6% - still with the
cubic symmetry intact.
