The structure of the poset of rational pointed cones

Let $\operatorname{\textsf{Cones}}(d)$ be the partially ordered set of pointed, rational, finitely generated cones in $\mathbb{R}^d$, ordered by successive addition of nonnegative integer multiples of lattice points. Let $\operatorname{\textsf{Cones}}(d)^+$ be the subposet of nonzero cones contained in $\big(\mathbb{R}^{d-1}\times\mathbb{R}_{>0}\big)\cup\{0\}$, and let $\operatorname{\textsf{Cones}}(d)^{(h)}$ be the corresponding height-filtered subposet. Cone-poset conjecture. For every $d,i,j$: (a) the order relation in $\operatorname{\textsf{Cones}}(d)$ is the inclusion order and therefore $|\operatorname{\textsf{Cones}}(d)^+|$ is contractible; (b) the relative groups

$$H_i\big(|\operatorname{\textsf{Cones}}(d)^{(j+1)}\setminus\{0\}|,|\operatorname{\textsf{Cones}}(d)^{(j)}\setminus\{0\}|\big)$$

are finitely generated $\mathbb{Z}[\operatorname{Aff}_{d-1}(\mathbb{Z})]$-modules. These assertions describe the expected homotopy and equivariant homological structure of the cone filtration; no resolution is supplied.