Finite ergodic decomposition conjecture for the piecewise affine square map

Let $F=F_\theta$ be the map of the square ${\mathcal S}$, with parameter $\theta\in(0,\frac{\pi}{4})$. An ergodic decomposition conjecture asserts that, for Lebesgue almost all $\theta\in(0,\frac{\pi}{4})$, there is a finite number of $F$-invariant sets $A_1,\ldots,A_m$, each of positive Lebesgue measure, such that

$$F\vert_{A_i}\colon A_i\to A_i$$

is ergodic for every $i=1,\ldots,m$. Each $A_i$ is a topological annulus with a certain number of elliptic islands removed from it, and, together with the elliptic islands and the invariant disk inscribed in ${\mathcal C}$, the $A_i$ form a partition of ${\mathcal S}$. This is presented as numerical evidence for typical parameters; the finiteness and precise topological description of the ergodic components remain unproved.