Benedetti–Guadagnini conjecture on CMC marked spectra

Let $M$ be a future complete $MGHC$ non-elementary flat space-time of dimension $2+1$, and let $T_{cmc}$ be its associated $CMC$ time. For each $a>0$, let $(S^{T_{cmc}}_a,d^{T_{cmc}}_a)$ be the metric space associated with the level set of $T_{cmc}$ at time $a$, and for $\gamma\in\pi_1(M)$ define its marked length by

$$l_a^{T_{cmc}}(\gamma)=\inf_{x\in S^{T_{cmc}}_a}d^{T_{cmc}}_a(x,\gamma.x).$$

Benedetti–Guadagnini conjecture. For every $\gamma\in\pi_1(M)$,

$$\lim_{a\rightarrow 0}l_a^{T_{cmc}}(\gamma)=l_{\Sigma}(\gamma),$$

and

$$\lim_{a\rightarrow +\infty}a^{-1}l_a^{T_{cmc}}(\gamma)=l_{\mathbb{H}^2}(\gamma).$$

The conjecture predicts that the CMC level-set marked spectra converge at the initial singularity to the spectrum $l_{\Sigma}$ and, after rescaling, at late times to the hyperbolic spectrum $l_{\mathbb{H}^2}$. The supplied text gives no resolution status.