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A map $f: K \to \mathbb{R}^d$ of a simplicial complex is an almost embeddingif $f(\sigma) \cap f(\tau) = \varnothing$ whenever $\sigma, \tau$ are disjointsimplices of $K$. Fix integers $d,k \geqslant 2$ such that $k+2 \leqslant d\leqslant\frac{3k}2+1$. Assuming that the "preimage of a cycle is a cycle" weprove $\mathbf{NP}$-hardness of the algorithmic problem of recognition ofalmost embeddability of finite $k$-dimensional complexes in $\mathbb{R}^d$.Assuming that $\mathbf{P} \ne \mathbf{NP}$ (and that the "preimage of a cycleis a cycle") we prove that the embedding obstruction is incomplete for$k$-dimensional complexes in $\mathbb{R}^d$ using configuration spaces. Ourproof generalizes the Skopenkov-Tancer proof of this result for $d =\frac{3k}{2} + 1$.

Hardness of almost embedding simplicial complexes in $\mathbb{R}^d$, II