Are nonrelativistic ground state ell-boson stars only stable for ell=0 and ell=1?

In previous work we analyzed the linear stability of non-relativistic $\ell$-boson stars with respect to radial modes and showed that ground state configurations are stable with respect to these modes, whereas excited states are unstable. In this work we extend the analysis to non-spherical linear mode perturbations. To this purpose, we expand the wave function in terms of tensor spherical harmonics which allows us to decouple the perturbation equations into a family of radial problems. By using a combination of analytic and numerical methods, we show that ground state configurations with $\ell > 1$ possess exponentially in time growing non-radial modes, whereas only oscillating modes are found for $\ell=0$ and $\ell=1$. This leads us to conjecture that nonrelativistic $\ell$-boson stars in their ground state are stable for $\ell=1$ as well as $\ell=0$, while ground state and excited configurations with $\ell > 1$ are unstable.