Necessity of the singleton-partition condition for omnivocal communication

Let $X_{[m]}=(X_1,\ldots,X_m)$ be a multiterminal source model with $m\geq 3$ terminals. Let omnivocal communication mean communication in which every terminal communicates, and let the singleton partition be the partition $\{\{1\},\ldots,\{m\}\}$; write $\mathbf{I}(X_{[m]})$ for the source-model partition functional and call a partition a minimizer when it attains the minimum of this functional.

Omnivocality conjecture. Omnivocal communication is necessary for achieving SK capacity if and only if the singleton partition is the unique minimizer for $\mathbf{I}(X_{[m]})$.

This conjecture proposes that the sufficient condition established in the paper exactly characterizes when every terminal must communicate to achieve secret-key capacity. The condition is known to be sufficient in the stated setting, while the paper reports that necessity is proved only for $m=3$; the general case remains open.