Generic contact Anosov flow conjecture for zero resonant states

Let $M$ be a compact $(2n+1)$-dimensional manifold and let $\alpha$ be a contact $1$-form on $M$ whose corresponding flow is Anosov with orientable stable and unstable bundles. Let $\operatorname{Res}^k_0$, for $0\leq k\leq 2n$, be the spaces defined by the cited resonant-state construction, and let

$$\pi_k:\operatorname{Res}^k_0\cap\ker d\longrightarrow H^k(M;\mathbb C)$$

be the corresponding map. Denote by $b_k(M)$ the $k$-th Betti number of $M$, and by $m_{\mathrm R}(0)$ the order of vanishing at zero of the Ruelle zeta function. Generic contact Anosov flow conjecture. For a generic choice of $\alpha$: the semisimplicity condition holds in every degree $k=0,\dots,2n$; $d(\operatorname{Res}^k_0)=0$ for every $k=0,\dots,2n$; and, for $k=0,\dots,n$, the map $\pi_k$ is onto and

$$\ker\pi_k=d\alpha\wedge \operatorname{Res}^{k-2}_0, \qquad \dim\ker\pi_k=\dim\operatorname{Res}^{k-2}_0.$$

Moreover,

$$\dim\operatorname{Res}^k_0=\sum_{j=0}^{\lfloor k/2\rfloor}b_{k-2j}(M),\quad 0\leq k\leq n; \qquad \dim\operatorname{Res}^{2n-k}_0=\dim\operatorname{Res}^k_0,$$

and

$$m_{\mathrm R}(0)=\sum_{k=0}^{2n}(-1)^{k+n}\dim\operatorname{Res}^k_0=\sum_{k=0}^n(-1)^{k+n}(n+1-k)b_k(M).$$

The conjecture predicts that generic perturbations destroy the extra symmetries responsible for non-closed resonant states in the hyperbolic case, making all zero-resonant states closed and determining their dimensions from the topology of $M$.