I forget the other papers, but there are other sources that discredit the HK.
Here's one By Louis Essen (the inventor of the atomic clock) that said the experiment was not nearly accurate enough...
L Essen, Electron. Wireless World 94 (1988) 238.
HK isn't really a concern anymore with the advent of GPS. GPS is far more accurate. If you're just looking into finding controversy, begin examining the GPS calculation adjustments made in jet airplanes. Accuracy there requires c+/-v calculations but I believe the mainstream explanation for this is simply the one-way sagnac effect.
Basically, the dissenter opinion is that to be accurate, jet airplanes must calculate the distance toward a sattellite they have traversed from the time the signal was transmitted to when it was received. They then subtract this distance. (this is actually true) By doing this, the end calculation of c is c+v.
On the other hand, this same apparent c+v happens in a sagnac device and the GPS satellites have proven to act in the same way as a sagnac ring such that the travel time of C in one direction around the Earth is faster than the other (simply a shorter distance) So the apparent problem with GPS on airplanes is just another example of the Sagnac effect already described by GR.
Either way, it makes for fun discussion. (and regardless of my beliefs, devil's advocate can be loads of fun)
Could someone correct me on the two opinions if I'm wrong...