Allais' spectrality conjecture for the contact spectral invariant

Let $(M,\ker\alpha)$ be a closed orderable contact manifold, and let $\widetilde{\operatorname{Cont}_0}(M,\ker\alpha)$ denote the universal cover of the identity component of its contactomorphism group. The map $c_+^\alpha$ is defined by the order relation with the Reeb flow. A map is $\alpha$-spectral when its value belongs to the appropriate spectrum associated with the contact isotopy. Allais' spectrality conjecture. If $(M,\ker\alpha)$ is a closed orderable contact manifold, then

$$c_+^\alpha: \widetilde{\operatorname{Cont}_0}(M,\ker\alpha)\to\mathbb{R}$$

is $\alpha$-spectral. This is the spectrality formulation used to obtain conjugation-invariance consequences for the rounded invariant; the other listed properties are known in the cited context, while this spectrality assertion remains open.