Connected $ω$-limit sets of Cournot maps: periodic orbits, realization and almost every point

Let $F(x,y)=(f_2(y),f_1(x))$ be a Cournot map of the unit square, $φ=f_2\circ f_1$ and $χ=f_1\circ f_2$. Linero Bas and Muñoz Guillermo proved that an $ω$-limit set of $F$ with nonempty interior is connected if and only if \textup{(C1)} its two projections are connected and \textup{(C2)} $ω_{F^2}(z)