Duality conjecture for ending lamination spaces and curve complexes

Let $S$ be a $p$-punctured surface of genus $g$. Write $\mathcal{C} (S)$ for its curve complex, $\mathcal{E} \mathcal{L} (S)$ for its ending lamination space, and $\mathcal{P} \mathcal{M} \mathcal{L} (S)$ for its projective measured lamination space. Let $h(\dim(\mathcal{C} (S)))$ denote the homological dimension of the curve complex. Duality conjecture. The following hold: (1) $\dim(\mathcal{E} \mathcal{L} (S))+h(\dim(\mathcal{C} (S)))+1=\dim(\mathcal{P} \mathcal{M} \mathcal{L} (S))$; if $\dim(\mathcal{P} \mathcal{M} \mathcal{L} (S))=2n+1$, then (2) for every $m$ there is a natural injection $H_m^{st}(\mathcal{E} \mathcal{L} (S))\to H^{2n-m}(\mathcal{C} (S))$, and (3) for every $m$ there is a natural injection $H_m(\mathcal{C} (S))\to \check H^{2n-m}(\mathcal{E} \mathcal{L} (S))$. The conjecture proposes a duality between the topology of ending lamination spaces and curve complexes; the source notes that Harer's computation makes part (1) equivalent to a predicted dimension formula, while parts (2) and (3) remain to be established.