Before revision:   We improve this by showing that if r is not a prime power and N:=(d+1)r-r\Big\lceild+2{r+1}\Big\rceil-2, then there is an almost r-embedding \Delta_N\to\mathbb R^d.
After revision:  This was improved by Blagojevi\'c-Frick-Ziegler using a simple construction of higher-dimensional counterexamples by taking k-fold join power of lower-dimensional ones. We improve this by showing that if r is not a prime power and N:=(d+1)r-r\Big\lceild+2{r+1}\Big\rceil-2, then there is an almost r-embedding \Delta_N\to\mathbb R^d.