  In [2], while studying a relevant class of polyominoes that tile the plane by
translation, i.e., double square polyominoes, the authors found that their
boundary words, encoded by the Freeman chain coding on a four letters alphabet,
have specific interesting properties that involve notions of combinatorics on
words such as palindromicity, periodicity and symmetry. Furthermore, they
defined a notion of reducibility on double squares using homologous morphisms,
so leading to a set of irreducible tile elements called prime double squares.
The authors, by inspecting the boundary words of the smallest prime double
squares, conjectured the strong property that no runs of two (or more)
consecutive equal letters are present there. In this paper, we prove such a
conjecture using combinatorics on words tools, and setting the path to the
definition of a fast generation algorithm and to the possibility of enumerating
the elements of this class w.r.t. standard parameters, as perimeter and area.
