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Yes, they do. That's where the extra (1 + z) factor in the definition of luminosity distance comes from.
I don't like this solution. There is nowhere the speed of light to be seen there.
OTOH, the "curvature of space" they mention, is not very necessary in our flat space.
But the Lorentz factor would be needed here. Not only for the time dilatation factor, by which the energy output is to be reduced - but also for the relativistic mass increase by the same factor. And for the length contraction as well!
That's the real problem, I think.