It's not about whether portals have mass or not. It's about the box's velocity relative to the space on the other side of the portal. In the game, as you move toward a portal, you also move toward the objects on the other side of the portal. Therefore you have velocity relative to those objects, and that's the momentum that is conserved as you pass through.

Lol. This will be fun. Glados' explanation is still correct, but momentum being conserved between portals (or "across portals" as I'd rather phrase it) is exactly what gets you answer B.

"A perspective that encompasses all of the objects involved" is not a thing in reality. No point in space cares about another point in space beyond the forces felt from that direction transmitted through spacetime, and the frame of reference following the left portal moving down and the frame of reference following the stationary right portal are both equally valid and moving relative to each other over the timeframe of the cube transitioning through the portal. I don't agree that the game is pushing a different model, but if it actually is, then that opens up so many (more) holes in physics that we may as well be debating magic.

I actually addressed the portal stopping halfway in another comment here which I won't bother to copy here, but the summary is that you shouldn't think of the box as one item but a mass of particles connected by internal forces. Your answer is indeed "obvious" and intuitive, but accelerating portals do funny things with frames of reference that means intuitive isn't always right and is the reason this topic always creates the disagreement it does.

As the portal comes down across the top half of the cube with constant relative velocity V and the appropriate momentum for said velocity, all those portaled particles will emerge from the other side of the portal with the same relative velocity V and same associated momentum compared to that side of the portal. Whether the particles maintain velocity relative to the portal because of something intrinsic to how space works at the portal boundary or because particles coming in forcibly displace particles going out in equal measure, the result is the same, fast in = fast out.

When the descending portal suddenly stops halfway, the particles already portaled through don't "know" this. They've still got the momentum of being sent through moments earlier reinforced by all the particles around them doing the same thing, so they "desire" to keep moving away from the portal. The particles of the bottom half of the box are similarly "ignorant" of all the shenanigans above and "want" to maintain the zero momentum they've had all along as measured by most of the objects in the scene.

This means the top and bottom halves of the box, despite never losing conservation of momentum, suddenly don't align with each other. The conflict of the top half "wanting" to keep moving away from the portal at velocity V and the bottom half "wanting" to stay where it is will resolve in one of two ways:

If the cube's internal forces are strong and "springy" enough to accommodate the disagreement, forces will propagate and equalize throughout the whole cube effectively averaging all "desired" motion out and the cube will fly out of the right portal at velocity ~1/2 V.

If the internal forces are not enough, the two halves of the cube will separate from each other, either cleanly at the portal boundary if the momentum disagreement massively overpowers internal forces or more messily and violently if it barely overpowers them. Obviously we don't have portals in real life to visualize this, but you can kind of get a feel for the effect through planetary bodies breaking up at Roche limits.

Edit: Wording the overall effect another way: Portals moving relative to each other effectively link two frames of reference to be one and the same over the portal boundary. If one portal changes velocity, the two frames of reference cease to align. This can cause conservation of momentum to appear to have been violated externally while actually being maintained in all the internal reference frames objects actually experienced.

If momentum is conserved then neither A or B is correct, the box should be an infinity thin sheet. If the portal is half way through the box if it maintains it's thickness the box has to be moving at the speed of the portal out of it.