Piecewise-affine factor-map conjecture for Lorenz-attractor perturbations

Let $\mathcal{R}$ be the collection of all maps $\mathcal{F}:[0,1]\to[0,1]$ that serve as factor maps for first-return maps $\psi_v:R\setminus W\to R$ of the Lorenz attractor, not necessarily only for parameters $v\in P$. For a map in $\mathcal{R}$, use a partition

$$0=c_0<c_1<\ldots<c_n<c_{n+1}=1$$

and let $\mathcal{I}$ be the invariant set in $[0,1]\setminus\{c_1,\ldots,c_n\}$. Piecewise-affine factor-map conjecture. Every $\mathcal{F}\in\mathcal{R}$ satisfies: (i) $\mathcal{F}$ is piecewise discontinuous, orientation-preserving and affine on each $(c_k,c_{k+1})$, with a jump discontinuity at every $c_i$; (ii) whenever $\psi_v$ is semi-conjugated to $\mathcal{F}$, there is an invariant set $I_v$ for $\psi_v$ and a continuous surjection $\pi:I_v\to\mathcal{I}$ such that

$$\pi\circ\psi_v=\mathcal{F}\circ\pi;$$

(iii) if $x\in\mathcal{I}$ has minimal period $n$ under $\mathcal{F}$, then $\pi^{-1}(x)$ contains a periodic orbit of minimal period $n$ for $\psi_v$. The conjecture is motivated by numerical Lorenz systems with heteroclinic knots other than trefoils, for which a more general piecewise-continuous model may replace the beta-transformations; no resolution is supplied.