This is the most powerful objection to the FE hypothesis I have encountered.
1. Are lines of longitude curved or straight? Straight, because (i) if we follow a path in the same direction as the Pole star, we must be travelling in a straight line (ii) every point on that line must have the same longitude.* Thus only the AE map is valid (the bipolar map has curved lines of longitude, e.g.).
2. But if the AE map is valid, the southern cross must appear in different directions depending on the longitude.
3. But it doesn’t. So, if the FE hypothesis is true, the AE map is the only valid representation of the world’s surface. But it is not valid, ergo etc.
I don't see any way of answering this objection.
*I define a line of longitude as the set of points where the time of high noon is identical.