Acyclicity and comparison conjectures for new globular and corner homology

Let $\mathcal{C}$ be an $\omega$-category, let $I^n$ denote the $n$-dimensional cube, let $A$ and $B$ be composable $n$-morphisms, and let $\widehat{\mathcal{C}}$ denote the modified $\omega$-category used to define the new globular homology. Write $H_*^{new-gl}$ for the new globular homology, and let $\square_n^\pm$ denote the corresponding corner construction. New-homology conjectures. The following assertions should hold: $H_p^{new-gl}(I^n)=0$ for every natural number $n$ and every $p>0$; for every $n\geqslant2$, if $t_{n-1}A=s_{n-1}B$, then $\square_n^\pm(A*_{n-1}B-A-B)$ is a boundary in the corresponding corner homology of $\widehat{\mathcal{C}}$; the canonical map

$$H_n^\pm(\mathcal{C})\longrightarrow H_n^\pm(\widehat{\mathcal{C}})$$

is an isomorphism for every $n\geqslant0$; and, as a consequence of the preceding conjectures, $h_n^\pm$ factorizes through the new globular homology theory. These claims are proposed as the properties needed to resolve the comparison problems for the new globular homology; the source gives no resolution.