Conjecture on lexicographic monotonicity of edges in exterior shifting

Let $K$ be a triangulation of a compact surface without boundary. Let $\Delta(K)$ be its exterior algebraic shifting, let $\Delta(K)_1$ denote its edges, and let $<_{\mathrm{lex}}$ be the lexicographic order. For edges $\sigma,\sigma'\in\Delta(K)_1$ with $\sigma<_{\mathrm{lex}}\sigma'$, consider whether adjoining vertex $1$ produces a face of $\Delta(K)$.

Lexicographic edge conjecture. If $\sigma<_{\mathrm{lex}}\sigma'$, then

$$\{1\}\cup\sigma\notin\Delta(K)\Rightarrow\{1\}\cup\sigma'\notin\Delta(K).$$

This would impose a monotonicity condition on the edges of the exterior shifting of every compact surface triangulation. The source gives no resolution, so the conjecture remains open.