Uniqueness of triangle colours within components of the 2-coloured flip graph

Let $P_{n+2}$ be a convex polygon, and consider its $2$-coloured flip graph, whose vertices are triangulations with triangles coloured using two colours. A triangle is understood by its three vertices and its position in the polygon. The triangle-position uniqueness conjecture. In any connected component of the $2$-coloured flip graph, a triangle cannot appear in the same position with different colours. The authors observed this in the examples they considered, including components displayed for polygons with $n=6$ and $n=7$, but the supplied text gives no proof or resolution.