  Plateau problems with elastic boundary energy have been of recent theoretical
and applied interest. However, avoidance of self-intersection on the boundary
during the gradient descent requires strong assumptions on the initial
conditions. To make progress towards an alternative to these strong
assumptions, we introduce the novel M\"obius-Plateau energy, where we instead
use the repulsive M\"obius energy first introduced by O'Hara, and later
expounded upon by Freedman, He, and Wang. This new energy also has myriad
scientific utility and presents interesting theoretical phenomena.
  We first prove the existence of minimizing curves in the case of immersed
discs bounded by knots, placing our work on solid footing. We then investigate
the M\"obius-Plateau energy of helicoidal strips, which are classified as
screws or ribbons based on the orientations of the boundary helices. Through
direct methods of the variational equation, we show that stable helicoidal
screws are plentiful, whilst stable helicoidal ribbons impose strong
constraints on their parameters: they must have high coiling, thin width in
comparison to the coiling, and remain close to axis.
