The balanced linkless-embeddability conjecture

Let $K$ be a $d$-dimensional balanced simplicial complex, and let $<$ be a $(3,\ldots,3)$-admissible order. Let $K^{b,<}$ denote the corresponding balanced shift. Balanced linkless-embeddability conjecture. If $K$ is linklessly embeddable in $\mathbb{R}^{2d+1}$, then $K^{b,<}$ does not contain $[4]^{*(d+1)}$. This generalizes the graph case motivated by the non-linkless embeddability of $[4]^{*(d+1)}$ in $\mathbb{R}^{2d+1}$.