Regularity of critical immersions for scale-invariant curvature energies in even dimensions

For immersions of a closed $2h$-dimensional manifold $Σ$ into $\mathbb R^m$, we consider curvature energies that are invariant under ambient isometries and dilations, with the principal term $\int_Σ \big|\nabla^{(h-1)} \vec{\mathrm{I\!I}}\big|_g^2\,d\text{vol}_g$ and lower-order submanifold invarian