Kolmogorov-Barzdin embedding conjecture for higher-dimensional simplicial complexes

Let $X$ be a $k$-dimensional simplicial complex with $N$ simplices, and suppose that each vertex lies in at most $L$ simplices. Let $n\ge 2k+1$. An embedding is 1-thick if the distance between the images of any two non-adjacent simplices is at least $1$. Higher-dimensional Kolmogorov-Barzdin conjecture. There is a 1-thick embedding of $X$ into the $n$-dimensional ball of radius

$$R\le C(n,L)N^{\frac{1}{n-k}}.$$

This conjecture seeks the sharp higher-dimensional analogue of the Kolmogorov-Barzdin estimates for thick embeddings of graphs. The paper establishes an upper bound with an additional factor of $N^{\epsilon}$ and proves near-sharpness up to such factors, so the conjectured removal of the $N^{\epsilon}$ loss remains open.