Conjugacy conjecture for Lorenz-attractor first-return maps

Let $P$ be the parameter space, let $R_0$ and $R_1$ be the two components of the cross-section $R$, and let $h_v:R_0\cup R_1\to\mathbb{R}^2$ be the map specified in Fig. S. For $v\in P$, write $\psi_v:R_0\cup R_1\to R$ for the first-return map of the attractor and consider its invariant set. Conjugacy conjecture. For every parameter $v\in P$, the restriction of $\psi_v$ to its invariant set is conjugate to the restriction of $h_v$ to the corresponding invariant set. This conjecture proposes a measurable/topological pullback of the model dynamics to the Lorenz attractor; the source gives no resolution, and the possible expansion of points to continua is identified as the main obstruction to dense-orbit and statistical conclusions.