The symmetric-shifting conjecture for 2-dimensional complexes embeddable in 4-space

Let $K$ be a $2$-dimensional simplicial complex, and let $K^s$ denote its symmetric algebraic shifting. Symmetric-shifting conjecture. If $K$ is embeddable in $\mathbb{R}^4$, then $K^s$ does not contain the $2$-face {5,6,7\}. This conjecture would imply the Kalai–Sarkaria upper bound with the sharp constant $C=4$.