A cosmologically redshifting puzzle in a box

In the comments for another blog post about cosmological distances, I suggested a thought experiment that would highlight how the redshift from distant galaxies is not due to a Doppler shift, that is, light from very distant galaxies is not redshifted because of the movement of the galaxies, but rather because the intervening space itself is expanding:

If someone managed to trap light when the universe first became transparent, about 300,000 years after the Big Bang, inside a perfectly reflecting mirrored box such that the light is contained in perpetuity, what would have happened to the wavelength of the trapped light by now?

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