Generic semisimplicity conjecture for time changes of transitive Anosov flows

Let $X$ be a transitive Anosov vector field on a closed 3-manifold $\mathcal{M}$. Let $\Omega_0^1$ denote the space of closed resonant $1$-forms at zero, and let $\mathcal{L}_{fX}$ be the Lie derivative along the time-changed vector field $fX$. Generic semisimplicity conjecture. There exists an open and dense set $\mathcal O\subset C^{\infty}(\mathcal{M}; \mathbb{R}_{>0})$ such that for $f\in \mathcal O$, the action of $\operatorname{\mathcal{L}}_{fX}$ on $\Omega_0^1$ is semisimple. The conjecture concerns the remaining obstruction to computing the order of vanishing at zero of the Ruelle zeta function from geometric multiplicities. Semisimplicity can depend on the parametrisation: some time changes of geodesic flows of hyperbolic surfaces are known not to be semisimple, while the generic assertion remains open.