Consider two $d$-dimensional convex polytopes $c_1, c_2$ that share a $(d-1)$-dimensional face $f$. Let $M$ be a surface ($2$-manifold) that intersects each of $c_1$ and $c_2$ in a $2$-ball. Suppose that $f$ is intersected in two $1$-balls; see Figure.

On each side of $f$ the two $1$-balls must connect to a $2$-ball in $c_1$ and $c_2$. Thus either these $2$-balls form a sphere or there must be a tunnel bounded by these $2$-balls. In the case where there is a tunnel, choose a loop $\gamma$ around that tunnel. Can I tighten $\gamma$ so that the resulting loop $\tilde{\gamma}$ lies in a 2-dimensional plane?