I feel qualified to comment on this subject. Why? Because for one, I own an ABS barrel and have for a couple of years. Also, my company owns an infrared cammera that I can use at will. I also work in an industry that affords me knowledge of conductivity of various materials.
I couldnt tell you if carbon barrels are stiffer or more flexible than steel barrels of equal demensions. One member here years ago claimed his test showed that steel was stiffer.
Harmonics. Couldnt tell you that either. What I can tell you is that the ABS barrel I have flat out shoots. It isnt picky either. 225's, 250's, 300's, Hornady, Nosler, Sierra, Berger it doesnt matter. I gave up trying to find the perfect charge weights. I load them to the velcoities I want and it shoots.
Heat conductivity I have proven to myself that the ABS barrel conducts heat out to the surface of the barrel MANY times faster than a steel barrel of identical demensions using identical loads with the same cartridge with the use of an infrared camera shooting each barrel side by side in the same lighting and temperature.
As far as thermal conductivity of carbon, it varies greatly on what type of carbon is used, how it is treated AND the method in which it is applied. The difference between ABS and Cristiensen is night and day. Any time you take fiber strands and weave them together they act like fiber optic cable. It works great for conduction but does little for transfer. It is hard for wrapped fiber to transfer heat from one strand to another.
That said, for the first year, I was not sure if the barrel was worth it. After using it for a couple of years and objectively comparing it to other rifles including my other rifles, I wouldnt trade it for anything. It is perfectly balanced and shoots like an absolute dream. When I wear the barrel out, I will order another just like it. It is overkill for things like sheep, antelope and deer. However with the raw accuracy I am getting, it has become my goto rifle.