Skopenkov's NP-hardness conjecture for geometric embeddability

For integers $k,d$, let $\operatorname{GEM}_{k \rightarrow d}$ denote the decision problem of determining whether a given $k$-complex has a geometric embedding in $\mathbb{R}^d$. Skopenkov's conjecture. The problem $\operatorname{GEM}_{k \rightarrow d}$ is NP-hard for all $k,d$ satisfying

$$\max\{3,k\} \leq d \leq \frac{3}{2}k+1.$$

Geometric embeddability is decidable for all $k,d$ because it can be expressed in the first-order theory of the reals. The conjectured NP-hardness extends the known complexity landscape for piecewise-linear embeddability, while the cited context identifies $\operatorname{GEM}_{2 \rightarrow 3}$ as an interesting open problem.