I'm going to outline my argument for theory B (the cube is launched).

First, let me summarize what I believe to be the "common sense" argument for each position.

Common sense argument for theory A (the cube is not launched): From the perspective of the cube, it is sitting on top of the platform; even if there is no gravity, it is in contact with the platform. With no external force acting upon it, why would it suddenly accelerate away from the platform?

Common sense argument for theory B (the cube is launched): When the cube is passing through the portal, an observer on the blue side of the portal would observe the cube having considerable momentum. If you put your hand above the portal as the cube exited, your hand would be knocked back. Without some external force to slow down the cube, why would it suddenly and instantaneously stop moving?

To get the right answer, I propose that we need only make one crucial observation:

"When a portal accelerates relative to you, everything on the other side of the portal accelerates equally relative to you."

To see that this is true, consider two opposite cases:

Case 1: You are looking at a stationary orange portal 5 m in front of you. On the blue side of the portal, there is a stationary apple 5 m away from the portal. Currently, the apple is 10 m away from you and has 0 velocity. The orange portal begins accelerating toward you until it is moving toward you at 1 m/s. Necessarily, the apple begins accelerating toward you equally, so that it is also moving toward you at 1 m/s.

Case 2: Imagine the opposite case. The orange portal is moving toward you at 1 m/s. The apple (which is stationary relative to the blue portal) is thus also moving toward you at 1 m/s. If the orange portal suddenly stops (i.e. decelerates or accelerates in the opposite direction), then the apple will also stop, exhibiting the same acceleration as the portal, relative to you.

Once we have recognized this as true, we see that theory B describes the correct behaviour (the cube is launched).

Initially, the cube has a velocity of 0 relative to the platform it is sitting on, and a non-zero velocity relative to everything on the blue side of the portal. The cube begins passing through the portal, and then once it is entirely through, the platforms collide and the orange portal decelerates (or accelerates in the opposite direction).

What should we expect to happen? From the perspective of the platform, we would expect to see everything on the blue side of the portal accelerate in the opposite direction. This is obviously true for everything that was initially on the blue side of the portal: it was originally moving closer to the platform because the orange portal was moving closer to the platform, but now the orange portal is no longer moving closer, and so everything that was originally on the blue side of the platform has stopped moving closer.

But what of the cube? Initially, the cube was stationary relative to the platform. So when the orange portal decelerates, the cube (which is now on the other side of the portal) would exhibit the same acceleration (relative to the platform) as everything else on the blue side of the portal. This acceleration in the opposite direction causes the cube to begin moving away from the platform.

The cube is launched.

A classic

Copied from the other post:

It would violate relativity if it was A, though right? For example, one of the important thought experiments of relativity is that if you are in a sealed room, you cannot identify whether or not the room is stationary or has some velocity.

But if portals worked like the way they do in A, couldn't you could devise an experiment? Suppose there are two scenarios, one where the room is moving with some velocity, let's say 1 m/s downwards that is orthogonal to the floor and ceiling. No gravity, no friction, etc. Then you can have two portals, one that is pointed orthogonal to the room's velocity (so, mounted on one of the walls), and another portal that is mounted on the ceiling. Then, hold a table with a companion cube on it like in the diagram in the post, and ram it upwards at a speed of 1 m/s so the cube goes through the portal in the ceiling.

Then if the room is moving, then the table you are holding is stationary, because the room is traveling downwards at 1 m/s and you are moving the table upwards at 1 m/s, so the cube would just plop out of the second portal with no velocity. But if the room was stationary, then the table you are holding is moving, so it would go out of the second portal at 1 m/s in an orthogonal direction. So you can tell if your room is moving or not based on that, whether or not the block plops out of comes out at velocity 1 m/s.

So it must be B, in both cases (table stationary and portal moving, or table moving and portal stationary) the cube must fly out at velocity v. This way, there is no difference. It could be interpreted as the moving portal transferring its momentum somehow to the block.

Of course, we are assuming principles of physics apply to fictional portal that does not exist, so this might just be wrong.

Momentum is conserved with respect to the reference frame. In the new reference frame, the cube is moving quickly. It has no momentum on the platform, and has significant momentum after passing through the portal with high velocity. The apparent lack of conservation of momentum is due to the two reference frames being different.

It's like there are two superimposed identical universes, but one is stationary wrt the cube, and the other is moving wrt the cube. In the first universe, the cube has no momentum. In the second universe, the cube has high momentum. "Falling into" the second universe out of the first one would make it look like the cube manifested with significant momentum, even though it was at rest in its original universe.

There is no actual "paradox," except in the sense that this result may be unintuitive. (It's like how one-boxing Newcomb is objectively correct, but the premise breaks some assumptions in some of the usually-applicable math due to the impossible conditions.)

You should add the image and clarify that Yes is "B" while No is "A".

@Shemetz Done!