The Lorenz renormalization conjecture on rigidity and renormalization alternatives

Let $\mathcal{T}_w$ be the set of infinitely $w$-renormalizable Lorenz maps. A rigidity class of $f\in\mathcal{T}_w$ is the set of maps in $\mathcal{T}_w$ smoothly conjugate to $f$ on their attractors. The successive renormalizations of $f$ are attracted to a degenerate flipping $2$-cycle when $\mathcal{R}^{2k}f$ and $\mathcal{R}^{2k+1}f$ converge to smooth maps on $[0,1]$, while their critical points satisfy $c(\mathcal{R}^{2k}f)\to0$ and $c(\mathcal{R}^{2k+1}f)\to1$, or vice versa. The Lorenz renormalization conjecture. For each $w$ such that $\mathcal{T}_w\neq\emptyset$, exactly one of the following holds, and conversely each alternative is realized by some $w$: (A) $\mathcal{T}_w$ is a rigidity class and the stable manifold of a hyperbolic renormalization fixed point; (B) $\mathcal{T}_w$ is foliated by codimension-$1$ rigidity classes, one of which is the stable manifold of a hyperbolic renormalization fixed point, and every $f\in\mathcal{T}_w$ outside this stable manifold has successive renormalizations attracted to a degenerate flipping $2$-cycle; or (C) there is a nonempty open connected set $\mathcal{T}_w^\star\subsetneq\mathcal{T}_w$ that is a rigidity class and the stable manifold of a hyperbolic renormalization fixed point, its complement has two connected components foliated by codimension-$1$ rigidity classes, the boundary of $\mathcal{T}_w^\star$ is a rigidity class and the stable manifold of a hyperbolic renormalization periodic point of strict period two, and every map in the complement outside this stable manifold has successive renormalizations attracted to a degenerate flipping $2$-cycle. The supplied text presents this as the paper's main conjecture; no resolution is given.