Let $\square=[0,1]\times[0,1]$ be the unit square and $f\colon\square\to \square$ is a continuous map that fixes the points on the boundary.

Assume $f$ is a limit of homeomorphisms $\square\to \square$. (By Moore's theorem it is equivalent to the condition that for any point $p\in \square$ the inverse image $f^{-1}\{p\}$ is connected and its complement is connected.)

Is it true that for most segments in $\square$, their inverse images are Jordan arcs?

Say, given $s\in [0,1]$ consider the vertical unit segment $I_s$ defined by $x=s$. Is it true that for a dense G-delta set of values $s\in [0,1]$ the inverse image $$J_s=f^{-1}I_s$$ is a Jordan arc?

**Comments:**

If $f$ is injective on $J_s$ then $J_s$ is a Jordan arc (evident). One may expect that $f$ is injective for most of values $s$, but this is not the case --- even if all $J_s$ are Jordan arcs, the map $f$ might map an arc in $J_s$ to one point for all $0<s<1$.

For sure $J_s$ has vanishing measure for all but countable set of values $s$.

The problem can be reformulated in terms of decomposition of $\square$ into the inverse images $f^{-1}\{x\}$. The main trouble comes from the sets $f^{-1}\{x\}$ that are not Jordan arcs nor single points. I do not know examples with uncountably many such sets. (I alos do not know if one can find uncountably may connected but not path connected compact disjoint sets in the plane.) These problems are related to the problem that there at most countable set of Y shapes in the plane, see "Ys in the plane" in "Mathematical Puzzles" of Peter Winker.