Unconditional and Conditional Density Forecast Calibration

I have been reading a new paper by Tilman Gneiting and Johannes Resin (GR2021) -- deep and thought-provoking as always -- "Regression Diagnostics meets Forecast Evaluation:  Conditional Calibration, Reliability Diagrams, and Coefficient of Determination."

Their paper pushes in a variety of interesting ways toward aspects of conditional as opposed to merely unconditional (uniform probability transform, or PIT) density forecast calibration.    

That's largely what the iid part of the Diebold-Gunther-Tay (DGT 1998) result ("correct conditional calibration implies PIT ~ iid U(0,1)") is about, and DGT emphasize and illustrate that one needs to check not just the condition for correct unconditional calibration (PIT ~ U(0,1)), but rather the joint condition PIT ~ iid U(0,1).  

I want to understand more about the GR2021 results in their relation to the iid part of the DGT1998 PIT ~ iid U(0,1) result.      

Some relevant history and perspective are at https://fxdiebold.blogspot.com/2018/06/peter-christoffersen-and-density.html.

The GR2021 paper and code are at:

https://urldefense.com/v3/__https://arxiv.org/abs/2108.03210__;!!IBzWLUs!BtUAeIBxhkYOch-aPxSnALk9MfYczGKu4G37CXTTHnimafIb85ov4cY12xVEYhicRaVT$

https://urldefense.com/v3/__https://github.com/resinj/replication_GR21__;!!IBzWLUs!BtUAeIBxhkYOch-aPxSnALk9MfYczGKu4G37CXTTHnimafIb85ov4cY12xVEYsNE2Kt1$