Structural stability characterization for piecewise Möbius transformations

Let $F$ be a piecewise Möbius transformation with component transformations $f_k$, and let structural stability, hyperbolicity, and $b1$-expansion have their meanings defined in the paper. Structural-stability characterization conjecture. If $F$ is a structurally stable PMT, then each component transformation $f_k$ is loxodromic, $F$ is hyperbolic, and $F$ is $b1$-expanding. The conjecture is motivated by experimental evidence and is stated as a stronger version of the preceding conjecture; no proof or disproof is supplied.