Oscillation-energy conjecture for translated points on compact contact manifolds

Let $(Y,\alpha)$ be a compact contact manifold, and let $\varphi$ be a contactomorphism of $Y$. The spectral gap of $\alpha$ is the minimal positive action of a Reeb orbit of $(Y,\alpha)$, and $|\varphi|^{\mathrm{osc}}_{\alpha}$ denotes the oscillation norm. The implication

$$|\varphi|^{\mathrm{osc}}_{\alpha}<2(\text{spectral gap of }\alpha)\implies \varphi\text{ has at least one translated point, with at least }\dim H_{*}(Y,\mathbb{Z}/2)\text{ in the case all the translated points are non-degenerate}$$

Oscillation-energy conjecture. If $\varphi$ is a contactomorphism of an arbitrary compact contact manifold $Y$, then the implication above holds.

Shelukhin established this implication under an exact-filling assumption on $Y$ using Rabinowitz Floer homology; the conjecture asks for the result without that assumption.