Corner homology of the tensor product with the interval

Let $\mathcal{C}$ be an $\omega$-category and let $I^1$ be the interval in the biclosed monoidal structure on $\omega\operatorname{Cat}$. For either choice of sign, consider the $\omega$-functor $\mathcal{C}\to I^1\otimes\mathcal{C}$ sending $u$ to $R(\mp)\otimes u$. Interval-tensor conjecture. This functor induces an isomorphism

$$H_*^\pm(\mathcal{C})\longrightarrow H_*^\pm(I^1\otimes\mathcal{C})$$

and, for $*>0$, induces the zero map

$$H_*^\mp(\mathcal{C})\longrightarrow H_*^\mp(I^1\otimes\mathcal{C}).$$

The source notes that the assertion is easy to verify in low dimensions and gives only a low-dimensional check for one of the zero maps; the general statement remains open.