I am trying to find out whether there exists a Borel set $F\subset[0,1]^2$ that is strictly self-similar and symmetric and such that $0<\lambda(F)<1$, where $\lambda$ is the Lebesgue measure. We say $F$ is strictly self-similar and symmetric if it is strictly self-similar in the sense of this definition and if it is symmetric in both diagonals of $[0,1]^2$. An example would be if, for example, every one of the smaller squares $F\cap[0,1/2]^2$, $F\cap[0,1/2]\times[1/2,1]$, $F\cap[1/2,1]\times[0,1/2]$ and $F\cap[1/2,1]^2$ are replicas of $F$. The Sierpinski carpet is an example of what I'm looking for, except that it has Lebesgue measure 0.
I suspect that such a set does not actually exist. I have looked at the following list of fractals and at rep-tiles, but there does not seem to be anything that satisfies exactly what I'm looking for. I also found this paper on an attractor of Cantor-type with positive Lebesgue measure, but I'm not sure if it is symmetric and if the set generalizes to the unit square. Does anybody have an example of a self-similar and symmetric set or, if one does not exist, a reference/proof?