  We establish a weak-strong uniqueness principle for the two-phase
Mullins-Sekerka equation in the plane: As long as a classical solution to the
evolution problem exists, any weak De Giorgi type varifold solution (see for
this notion the recent work of Stinson and the second author, Arch. Ration.
Mech. Anal. 248, 8, 2024) must coincide with it. In particular, in the absence
of geometric singularities such weak solutions do not introduce a mechanism for
(unphysical) non-uniqueness. We also derive a stability estimate with respect
to changes in the data. Our method is based on the notion of relative entropies
for interface evolution problems, a reduction argument to a perturbative graph
setting (which is the only step in our argument exploiting in an essential way
the planar setting), and a stability analysis in this perturbative regime
relying crucially on the gradient flow structure of the Mullins-Sekerka
equation.
