Wagner–author non-embeddability conjecture for skeletons with a missing face

Let $K$ be a triangulated $2d$-sphere and let $T$ be a missing $d$-face in $K$, meaning that all proper faces of $T$ belong to $K$ but $T$ does not. Set

$$L=\operatorname{skel}_d(K)\cup\{T\}.$$

Non-embeddability conjecture. The complex $L$ does not embed in $\mathbb{R}^{2d}$. This generalizes the example discussed in the source and is described there as joint work in progress of Uli Wagner and the author; no proof or resolution is given.