The PL-embeddability form of the balanced Kalai–Sarkaria conjecture

Let $K$ be a $d$-dimensional balanced complex, and let $<$ be a $(2,\ldots,2)$-admissible order. Let $K^{b,<}$ denote the corresponding balanced shift. Balanced PL-embeddability conjecture. If $K$ is topologically embeddable in $\mathbb{S}^{2d}$, then $K^{b,<}$ is PL embeddable in $\mathbb{S}^{2d}$. This is stated as an equivalent formulation of the balanced Kalai–Sarkaria conjecture, using that $[3]^{*(d+1)}$ is not PL embeddable in $\mathbb{S}^{2d}$.