The balanced shifting conjecture for the van Kampen obstruction

Let $K$ be a $d$-dimensional balanced complex, and let $\mathfrak{o}(K)$ denote its van Kampen obstruction to PL embeddability in $\mathbb{S}^{2d}$. Balanced van Kampen-obstruction conjecture. If

$$\mathfrak{o}(K)=0,$$

then

$$\mathfrak{o}(K^b)=0.$$

For $d\geq 3$, the source explains that this conjecture implies the balanced Kalai–Sarkaria PL-embeddability formulation, including with $\mathbb{Z}_2$ coefficients.