The missing-face embeddability conjecture for triangulated spheres

Let $\mathsf{S}_{\leq d}$ be the $d$-skeleton of a triangulated $2d$-sphere and let $M$ be a missing $d$-face of $\mathsf{S}_{\leq d}$. Then $\mathsf{K}:=\mathsf{S}_{\leq d}\cup\{M\}$ is the resulting complex.

Missing-face embeddability conjecture. The complex $\mathsf{K}$ does not embed in the $2d$-sphere.

The theorem preceding this conjecture establishes the same assertion for piecewise linear spheres. The conjecture claims that the piecewise-linearity hypothesis is unnecessary.