Uniqueness of worst-case clusterings under a generalized exclusive-element condition

Let ${\cal C}$ be a given clustering, and let ${\cal N}_i$ be the set of elements belonging to the $i$th cluster. For every ordered subset of $\Delta+1$ distinct clusters $p_1,p_2,\dots,p_{\Delta+1}\in[k]$, assume that

$${\cal N}_{p_1}\setminus\left\{\bigcup_{p_j\neq p_1}{\cal N}_{p_j}\right\}\neq\varnothing.$$

Generalized uniqueness conjecture. The ground-truth clustering ${\cal C}$ is the only valid clustering consistent with the entire query matrix.

This conjecture is presented as the natural extension of the paper's result for $\Delta=2$ to arbitrary $\Delta>0$. Its status is not resolved in the supplied text.