The symmetric shifting obstruction to embeddability in even-dimensional spheres

Let $\mathsf{K}$ be a simplicial complex, let $\Delta^s(\mathsf{K})$ denote its symmetric algebraic shifting, and let $d$ be a nonnegative integer. Consider the set $\{d+3,d+4,\ldots,2d+3\}$ and the shifted-complex condition $\Delta_{\leq d}^{2d+2}\subseteq\Delta^s(\mathsf{K})$.

Symmetric shifting obstruction conjecture. If

$$\{d+3,d+4,\ldots,2d+3\}\in\Delta^s(\mathsf{K}),$$

equivalently, by shiftedness, if $\Delta_{\leq d}^{2d+2}\subseteq\Delta^s(\mathsf{K})$, then $\mathsf{K}$ does not embed in the $2d$-sphere.

This is presented as the part of the Kalai–Sarkaria conjecture sufficient, together with Swartz's result, to establish the algebraic $g$-conjecture for PL spheres. Its resolution is not supplied in the source.