Corner homology of cubes, free morphism categories and globes

Let $I^n$ be the $n$-cube, let $2_n$ be the free $\omega$-category generated by an $n$-morphism, and let $G_n$ for $n>0$ be the oriented $n$-globe, namely the free $\omega$-category generated by two non-homotopic $n$-morphisms with the same $(n-1)$-source and $(n-1)$-target. Basic corner-homology conjectures. For $p>0$, $H_p^\pm(I^n)=0$; if $p>0$, then $H_p^\pm(2_n)=0$; and for $p>0$ with $p\neq n$, $H_p^\pm(G_n)=0$, while

$$H_n^\pm(G_n)=\mathbb{Z}.$$

These are stated as elementary test cases for the corner homology theories and are said to be verifiable in lower dimensions; no general proof is supplied.