Characteristic-zero exterior shifting conjecture for surface triangulations

Let $K$ be a triangulation on $n$ vertices of a connected compact surface without boundary, and let $\Delta^{ex}$ denote exterior algebraic shifting over a field of characteristic zero.

Characteristic-zero exterior shifting conjecture.

$$\{1,3,n\}\in\Delta^{ex}(K).$$

This conjecture is stated as a sufficient condition for the barycentric subdivision conjecture. The source gives no resolution; it attributes the related rigidity context to Bulavka (2023).