On Simplices in Diameter Graphs in $\mathbb R^4$

A graph $G$ is a diameter graph in $\mathbb R^d$ if its vertex set is a finite subset in $\mathbb R^d$ of diameter $1$ and edges join pairs of vertices a unit distance apart. It is shown that if a diameter graph $G$ in $\mathbb R^4$ contains the complete subgraph $K$ on five vertices, then any triangle in $G$ shares a vertex with $K$. The geometric interpretation of this statement is as follows. Given any regular unit simplex on five vertices and any regular unit triangle in $\mathbb R^4$, then either the simplex and the triangle have a common vertex or the diameter of the union of their vertex sets is strictly greater than $1$.