Thin corner-cycle conjecture for free globular omega-categories

Let $\mathcal{C}$ be a free globular $\omega$-category. For a natural number $n$, let $x_i\in\omega\operatorname{Cat}(I^n,\mathcal{C})^\pm$ and let $\lambda_i$ be natural numbers indexed by a set $I$. Suppose that $x_i(0_n)$ has dimension strictly lower than $n$ for every $i$. Thin corner-cycle conjecture. The linear combination

$$\sum_i\lambda_i x_i$$

is a boundary if and only if it is a cycle. This would characterize boundaries among the thin elements of the corner complexes; the source gives no proof or resolution.