Existence of a tight point in every extreme configuration

The elements of a configuration are points in the $(m-1)$-dimensional probability simplex in $\mathbb{R}^m$. A configuration is extremal if it arises as the configuration associated with an extreme matrix $Q\in\mathcal{P}_{\delta}$. A point of a configuration is tight if it is at the maximal distance $\delta$ from every other point. Tight-point conjecture. There exists a tight point for all extreme configurations. This concerns the structure of extreme locally differentially private mechanisms and is stated in the setting where extreme matrices are being characterized for input sizes $k=2,3$ and arbitrary output size $m$; its resolution is not specified in the source.