Edge-maximizing projective transformation on polytopes

Let P and Q be simple polytopes such that P = Q ∩ H and let H be a halfspace with normal vector n. Let proj_n(e) denote the length of the projection of edge e onto vector n.

Consider the set E of edges of Q that cross through H, ie, edges with one endpoint in H and one outside H.

For any e ∈ E, does there always exist a projective transformation φ of P such that for every f ∈ E, proj_φ(n)(φ(e)) ≥ proj_φ(n)(φ(f))?

In other words, is there always a projective transformation making an arbitrary edge crossing H maximal with respect to H?

My first approach would be to choose φ to extend the endpoint of e as far as possible, but this does not appear to be sufficient to guarantee that proj_φ(n)(φ(e)) is maximized without adjusting other edges of Q.