The balanced-shifting embeddability conjecture

Let $K$ be a $d$-dimensional balanced simplicial complex, meaning that its vertices are colored with $d+1$ colors so that every edge has vertices of different colors. Let $K^b$ denote its balanced shifting. Balanced-shifting embeddability conjecture. Balanced shifting for $d$-dimensional balanced complexes preserves embeddability in $\mathbb{R}^{2d}$. The case $d=1$ is established by the paper's planar bipartite graph theorem; higher-dimensional cases remain conjectural.