  For an admissible class of smooth compact initial data sets with boundary, we
prove a comparison theorem between the Wang/Liu-Yau quasi-local mass of the
boundary and the Hawking mass of strictly minimizing hulls in the Jang graphs
of the domain. Using this, we prove a quasi-local Penrose inequality that
involves these quasi-local masses of the boundary and the area of an outermost
marginally outer trapped surface (MOTS) in the domain or the area of minimizing
minimal surface within the Jang graphs. Moreover, we obtain sufficient
conditions for the (non)existence of a MOTS within a domain, in the spirit of
the folklore hoop conjecture.
