Kalai–Sarkaria conjecture on algebraic shifting and embeddability

Let $K$ be a simplicial complex on $n$ vertices. For a shifted complex $\Delta(K)$ associated with $K$, let $\Delta(r+1,n)$ denote the pure $r$-dimensional simplicial complex defined by the maximal simplices satisfying the stated cyclic-polytope condition.

Kalai–Sarkaria conjecture. If $K$ admits an embedding into $S^{r}$, then

$$\Delta(K) \subseteq \Delta(r+1,n).$$

The type of algebraic shifting is not specified in the cited formulation. Since shifting preserves $f$-vectors, this containment would imply upper bounds for the face numbers of complexes embeddable in spheres and Euclidean spaces.