The balanced Kalai–Sarkaria conjecture

Let $K$ be a $d$-dimensional balanced complex, and let $<$ be a $(2,2,\ldots,2)$-admissible order, meaning that the least two vertices of each color class form an initial segment of the order. Let $K^{b,<}$ denote the corresponding balanced shift. Balanced Kalai–Sarkaria conjecture. If $K$ is topologically embeddable in $\mathbb{S}^{2d}$, then $K^{b,<}$ does not contain the van Kampen complex $[3]^{*(d+1)}$, the $(d+1)$-fold join of three points. The conjecture is presented as a balanced analogue of the “in particular” part of the Kalai–Sarkaria conjecture.