Gramination conjecture for the stretch locus

Let $G=\mathrm{PO}(n,1)$, let $\Gamma_0$ be the group in the representation space under consideration, and let $(j,\rho)\in\operatorname{Hom}(\Gamma_0,G)^2$. For $n=2$, assume that $j$ is geometrically finite, and let $E(j,\rho)$ be the intersection of the stretch loci of all $(j,\rho)$-equivariant maps with minimal Lipschitz constant $C(j,\rho)\in(0,1)$. Set $M:=j(\Gamma_0)\backslash\mathbb{H}^2$. Gramination conjecture. The set $E(j,\rho)$ is the lift to $\mathbb{H}^2$ of a gramination of $M$, meaning the union of a finite set $F$ and a lamination in $M\smallsetminus F$ with finitely many leaves terminating on $F$. This conjecture describes the still-mysterious stretch locus in the case $C(j,\rho)<1$; the paper gives more precise results when $C(j,\rho)=1$, but does not resolve this claim.