Parallel universe portals: an infinite hall of mirrors?

Parallel-Universe Portals

A solid material has been discovered that, when shaped into a ring and activated (by lowering its temperature below some critical point, let's say), it forms a parallel universe on the other side. This isn't a gateway into an existing alternate universe, but rather a duplicate of the host universe, identical in every way at the time of activation. However, over time (on the order of minutes), the two universes will desync due to slight quantum-scale differences that the portal produces on each side. (It's the same type of parallel universe generation described in this previous Worldbuilding SE question, but I'm interested in a special case)

As long as a portal is active, it must be in the same position in both universes. If you move the portal from one end, it moves in the other. This means that a portal can never be used to cross distance or time. When I say that it is in the same position, I also mean that it faces the same direction. So if you walk into a portal, moving north, you will exit in another universe, still moving north. Walk south again? Back in your universe, moving south. Walk around the portal and walk north through it again? Back in your universe, moving north this time. (Moving through the portals isn't really relevant to this question, but I just wanted to give some examples so that what I'm describing is a bit clearer.)

Notably, portals duplicate existing portals. To illustrate this, imagine that you have an active portal linking your universe, universe 0, to another universe, 1.

After a few minutes, when the universes are desynced, your copy in universe 1 decides to open another portal (since it's after the desync, you're able to choose not to open another portal in your universe). The end result is shown below, with 2 copies (joined by a blue portal) of your previous 2-universe system (joined by a red portal):

The Question

What happens if your copy doesn't wait for a desync before opening the second portal? Suppose once the red portal opens, she waits only 1 second before opening the blue portal. Since you're both still synced, you'll open a portal at that time as well. So there are 8 universes, right? I don't think so.

I think what happens is a "hall of mirrors effect." Consider the two copies of you at the ends of this 8-universe chain. Why do they only have a red portal open? Shouldn't they have also opened blue portals, since that's what you did and they're synced to you? By induction, the chain of universes should be infinite.

The question is: does the "hall of mirrors effect" produce a truly infinite chain of universes, or does something break the chain?

My current answer: the chain is infinite.

Reasoning: There's some incredibly slight amount of desync that happens in that 1 second between the portals opening. If a limit exists, it probably depends on that. Let's assume that 1 second of waiting causes a desync of 1 nanosecond (that is, that 1 second of waiting between portals cause you and your copy to behave almost the same, but with one of you on a 1-nanosecond lag). So the second blue portal opens 1 ns after the first.

I initially thought that this would make the chain finite, since each layer further from the starting layer would add time, until eventually the desync would be so large that the copies on the ends would be in a completely different state of mind and the hall of mirrors would be broken. But no, between 0 and 1 ns apart, the multiverse would resemble the second diagram, with 00 and 01 about to open their blue portals at the same time at t = 1 ns. Any role the tiny desync plays in the chain is gone by the first iteration.

Am I right? I'm not as sure about my answer as I'd like to be, and it feels a little wrong to have an actual infinity come up in this problem. After all, these portals are remarkably well-behaved in every other scenario I could think of. You can move them through each other and intersect them like chain links, all without making the multiverse implode or anything. I'm interested to see if this case has some subtlety I'm missing.

Answers could involve:

• The desync between the portals
• The time that light takes to travel between the portals
• The absurdly small probability of a quantum fluctuation causing, say, the portal to stop working, on a human timescale

Answers should not involve:

• Handwavium / built-in stopping mechanisms (i.e. the portals are designed not to work if another portal was opened less than a second earlier)
• The portals causing the universe to collapse (for General Relativity reasons or otherwise)
• Any assertion that the chain of universes doesn't exist at all (or at least think about it a while and draw some diagrams before posting. This is very easy to misunderstand. After a lot of thought, I'm confident that the chain starts, just not that it ends.)