Regardless 2D or 3D...

If you step outside right now and look West the crescent Moon with a very big and bright Venus above it and the faint red dot of Mars a little above Venus is a beautiful sight.

...step in front of a moving bus and see where that 2-dimensionality gets you.

Stepping in front of a bus has the ability to convert 3D matter to 2D matter.

a 2-D outer surface gets mapped to the 3-D interior... assuming it even happens!

It's clearly not proposing that the 3-D interior of our universe would be like a photographic hologram that isn't tangible.

It appears to be a mathematical construct that was proposed after examining principles related to black holes and recognizing it could be applied to the everything else in the universe as well. The holographic model wasn't pulled out of thin air, but was derived from efforts to avoid contradictions of basic principles of physics in regard to black holes.

How it works, assuming it's an accurate model for everything, isn't well understood. The holographic model makes some different predictions, and that's what is being checked at this point.

For example, there's subtle differences in predictions related to something called cosmic inflation. That's something that supposedly happened in the very early universe that helps explain the almost uniform temperature of the cosmic microwave background radiation. It's a mathematical model that was created to explain the CMB observations. In the holographic model, cosmic inflation is a natural consequence but the effects should be slightly different in the CMB.

Something else that's interesting is that Einstein's General Relativity equations can be derived by first creating a holographic model based on quantum theory! Is it the long-sought merger between quantum theory and relativity? That's sometimes called quantum gravity.

Here's a talk by Tom Banks that delves into it without a bunch of math (skip to about the 45-minute mark for some of the derivations based on the model):

Concepts like Gauss' Law and center of mass don't admit an exact distribution of matter. Maybe moments of distributions (probabilities?) could provide some information, which might shed light (sorry for the pun) on a question like why there are regions of dark matter. Just a guess.

When the matter-density in the universe dropped so low that photons could finally travel a significant distance without immediately running into another particle. We already had atoms at that point.
The CMB is an image what the universe looked like at that point... Hundreds of thousands of years after the Big Bang.

The Cosmic Neutrino Background was created a few seconds after the Big Bang, when the energy-density dropped so low that neutrinos could finally travel a significant distance without immediately running into another particle. We already had Quarks, Gluons and Leptons at that point.
If we could measure that more precisely, we could get an image of what the universe looked like at that point... Seconds after the Big Bang.

And even seconds after the Big Bang is already way, way, way, way too late to measure anything about the particle-physics right during the Big Bang.
The guys in the paper want to go to a time where even Quarks, Gluons and Leptons didn't exist yet.



Here's their conclusion:
We showed that holographic models based on three-dimensional perturbative QFT are capable of explaining the CMB data and are competitive to the ?CDM model. However, at very low multipoles (roughly l<30), the perturbative expansion breaks down, and in this regime the prediction of the theory cannot be trusted.

Translation:
The theory can predict the fine details of the CMB but not the larger general shape.

I climbed out and walked away thinking a two dimensional holographic universe was silly.

Humans often go wrong in our theorizing about the universe because of the way we experience time. Personally, I don't believe in time; time itself is an aspect of energy; the perception of time was a useful shortcut in the evolution of life. If you want to see how that might be, consider the odds of this: Every one of your ancestors from the beginning of life on earth managed to survive and reproduce. And here you are. Your time is now. (And hell no, I'm not being metaphysical here. I'm quite comfortable with "perception" as a physical process. A chemical process. A yeast cell can have a basic sort of perception.)

From the imaginary perspective of a photon there is no time. And then you've got Einstein's infamous equation E=mc². That doesn't mean mass can be turned into energy, or vice versa, it means mass *is* energy and energy *is* mass.

With that mental model I see there's only one speed in the universe, and that is the speed of light. Everything else is an interference pattern written upon that light.

That's how you end up with a holographic universe. It's not two dimensions and then this funny third dimension time. It's three identical dimensions. If, mathematically, you lock one of these dimensions down and transform the entire 3D universe across the other two dimensions, then you are not seeing the entire picture.