Ben Efraim's lexicographic edge-isoperimetric conjecture for the transposition graph

Let $S_n$ be the symmetric group, and let $T_n$ be its transposition graph. For $\mathcal{A} \subset S_n$, write $\partial \mathcal{A}$ for its edge-boundary in $T_n$. Let $\mathcal{C}$ be the initial segment of the lexicographic order on $S_n$ having size $|\mathcal{A}|$. Ben Efraim's conjecture. For every $\mathcal{A} \subset S_n$,

$$|\partial \mathcal{A}| \geq |\partial \mathcal{C}|.$$

This would give an edge-isoperimetric inequality for the transposition graph that is sharp for every set size; the source provides no resolution, so the conjecture remains open.