Sandon's small oscillation conjecture for translated points

Let $(N,\xi)$ be a contact manifold with a global contact form $\alpha$, and let $\psi$ be a contactomorphism. Say that $\psi$ has small oscillation energy when its oscillation energy is sufficiently small, and call $\psi$ non-degenerate when its translated points are non-degenerate. Sandon's small oscillation conjecture. Every contactomorphism $\psi$ of small oscillation energy has at least one translated point, and if in addition $\psi$ is non-degenerate, then it must have at least

$$\dim H_*(N,\mathbb{Z}/(2))$$

translated points. The result is proved under the additional hypothesis that $(N,\xi)$ admits a strong exact symplectic filling; removing that filling assumption is the proposed conjectural extension and remains open.