The permutation-degree conjecture for friendly triangulations

Let $T$ be a triangulation of the partition simplex $\Delta^{n-1}$, and let $L:\operatorname{Vert}(T)\to 2^{[n]}$ be a consistent labeling of $T$. A triangulation is friendly when it has the ownership structure required for the labeling $L$ in the paper, and a labeling is single-valued when it assigns one element of $[n]$ to each vertex.

Permutation-degree conjecture. The labeling $L$ induces a single-valued labeling $\ell:\operatorname{Vert}(T)\to [n]$ such that

$$\operatorname{bdeg}(\ell)\not\equiv 0\pmod n.$$

This conjecture is the extension to $n>3$ of the lemma used to prove the existence of connected envy-free divisions for three selective agents. Establishing it would extend the combinatorial argument to more agents; the paper states that the authors could not prove it for $n>3$.