Homological and cohomological inclusion conjecture for curve-complex models

Let $S$ be a finite-type surface. Let $\mathcal{C} (S)_{\operatorname{sub}}$ be the subspace model of the curve complex, and let $\mathcal{U} \mathcal{P} \mathcal{M} \mathcal{L} (S)_{\operatorname{cw}}$ and $\mathcal{U} \mathcal{P} \mathcal{M} \mathcal{L} (S)$ denote the CW-topologized and original versions of the unmeasured projective measured lamination space, respectively. Homological inclusion conjecture. The bijective maps $I:\mathcal{C} (S)\to\mathcal{C} (S)_{\operatorname{sub}}$ and $I:\mathcal{U} \mathcal{P} \mathcal{M} \mathcal{L} (S)_{\operatorname{cw}}\to\mathcal{U} \mathcal{P} \mathcal{M} \mathcal{L} (S)$ induce inclusions in Steenrod homology and Čech cohomology. The conjecture is presented as a step toward parts (2) and (3) of the duality conjecture; the source gives no resolution.