Thin-element decomposition conjecture for homotopic functors

Let $\mathcal{C}$ and $\mathcal{D}$ be free $\omega$-categories, let $f,g:\mathcal{C}\to\mathcal{D}$ be homotopic non $1$-contracting $\omega$-functors, and let $x\in\omega\operatorname{Cat}(I^n,\mathcal{C})^\pm$. Thin-element decomposition conjecture. There exist a boundary $B$ of $\mathbb{Z}\omega\operatorname{Cat}(I^n,\mathcal{D})^\pm$ and a linear combination $T$ of thin elements of $\mathbb{Z}\omega\operatorname{Cat}(I^n,\mathcal{D})^\pm$ such that

$$f(x)-g(x)=B+T.$$

This decomposition is presented as the additional fact needed to derive corner-homology invariance from the thin-element conjecture; it remains unresolved in the source.