I haven't read the paper, just the abstract. But the literal answer to your question is: You don't observe the cat. Ergo, it's like the "box" was never even opened.
Now that I've answered that, let's caution you about your deployment of Mr. Cat. First of all, I don't believe you'll need an uncertain quantum object to observe in this particular thought experiment. Life is complicated enough without bringing in more quantum mechanics than you need. Just imagine a brick. It's red. Light is shining on the brick and it looks red. Now, I install my time-bending lenses around the brick, and then at one particular instant the brick changes to be yellow, and then goes back to being red. But you don't see that, you just see a continuously red brick, even though the light was "shining on" the brick the whole time, because our lenses made a "time hole" in the stream of light and the color change happened during the hole.
I should also warn you that our friend Mr. Cat is a bad metaphor in many ways. (He was originally an in-joke, after all, used by people who were debating the apparent absurdity of a new theory.) The problem with the cat is that he's a completely unrealistic creature, and he primes your intuition in all sorts of incorrect ways. In real life you will never stumble across a coherent superposition of two quantum states as dramatically large (trillions of cells!) and dramatically different (alive vs dead!) as this imaginary cat. Real cats interact with their surroundings. A lot. They breathe and meow, they emit organic molecules that smell like cat, they charge up with static electricity (electric fields) and then run around in circles (magnetic fields), and above all they gravitate (which is difficult to cloak). So to make the thought experiment work the cat's imaginary box has to be a cloaking device, a box that permits no interaction with the outside world until it is "opened". And that doesn't exist in real life.
And, therefore, importing the cat into this thought experiment is a particularly bad move, because it's redundant. If you have Shroedinger's imaginary box you don't need an additional set of imaginary time-cloaking lenses. The box is already cloaking the cat, for a fairly long period of time.
Always be careful: Thinking about completely impossible things is fun for a while, but don't make a habit of it, or you'll become the sort of insufferable spaced-out quantum-mechanics groupie who says things like gosh, maybe when I turn around and stop believing in this tree it will cease to exist. Of course not. Tree-sized objects do not physically disappear from your universe, not without the sound of a chainsaw.