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I'll chime in here with anecdotal evidence since what little data I have is not from ideal methodology nor immediately at hand.
In my experience the best option is a "streamlined" bed cap, next to that is a flat tonneau, and next to that is tailgate up. This is with a Quad Cab/Short Bed combination.
I have no data but experience from towing open utility trailers has given me the hypothesis that the tailgate condition is more important on longer beds than on short bed pickups. It is possible that with a regular cab/long bed truck tailgate down may be slightly better than tailgate up, but I still cannot see how either could be better than a tonneau or tapered cap.
When my truck was only a few months old I took a highway trip from MD to GA down I-95 most of the way with a combination of our "new couple" holiday gift haul and moving some last items out of our parents' homes. My truck bed was loaded cab-high at the rear window and tailgate-high at the gate, covered in a blue tarp and bed net. The load was bed-wide at the rear window but got narrower (still gate-high) toward the gate. Since the net made the tarp hug the load, this approximated the "ideal" aero bed cap. Without trying, heavily loaded, with the cruise set at 75mph in a truck that was not completely broken in I pulled high-19s mpg when my previous best highway trip was 17s. I haven't been able to replicate this bed condition.
About two years after this I bought a soft tonneau cover and saw an average increase of 1-1.5 mpg IIRC, comparing long-term averages before and after but not A-B-A.
Towing a 16' utility trailer with 3' high wooden sides and solid 3' high tailgate I could barely pull past 65mph and was seeing instant readouts of 10-12mpg on flat ground (unloaded), so I shouldn't have been towing that in 5th. I stopped and yanked the gate and strapped it to the front floor of the trailer and could easily pull 70 in 5th with flat-ground instant readouts of 14-15 mpg. I think the trailer was long enough for the bubble of circulating air to end and the flow to approximate laminar on the trailer floor. This may happen in an 8' truck bed but likely will not in a 6' truck bed.
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