Regular simplex partition number formula

For $r\geq 3$ and $d\geq r-1$, let $N_{(\Delta_{r-1};d)}$ denote the least number of vertices needed so that every map $f:\Delta_{N-1}\rightarrow\mathbb{R}^d$ admits a partition into $r$ pairwise disjoint faces whose images form a regular $(r-1)$-simplex. Regular simplex partition formula.

$$N_{(\Delta_{r-1};d)}=\binom{r+1}{2}-1.$$

This exact formula is stated as a conjecture candidate by the parser, but the supplied context does not identify it as an open conjecture or provide resolution evidence; its status should therefore be checked against the paper's surrounding theorem and definitions.