Schur's conjecture on intersecting unit simplices

Let $S_1$ and $S_2$ be two unit simplices in $\mathbb R^d$ forming a set of diameter $1$, with $k$ and $m$ vertices respectively, and assume $k\geq m$. Schur's conjecture. The simplices $S_1$ and $S_2$ share at least

$$\min\{0,k+2m-2d-2\}$$

vertices. The conjecture arose in the study of cliques in diameter graphs and was confirmed in the two special cases $(k,m,d)=(d,d,d)$ for $d\geq2$ and $(k,m,d)=(5,3,4)$, but its general resolution status is not established by the supplied evidence.