The thermal conductivities of Pure Nitrogen and Pure Oxygen are not too much different, but it is true as a general rule that the thermal conductivity of a single pure gas is somewhat higher than when another gas is mixed in. A mixture of two gases, such as nitrogen and oxygen, would be expected to have a "cooling ability" less than the better of the two, and in many cases less than either of the two alone.

Note that the definition of "general rule" is "anything that works except when you rely on it."

The use of pure, or nearly pure, nitrogen for racing, based on improved tire heat dissipation thus has some credibility; and it's quite likely that a search would turn up some fairly persuasive accurate tests that will confirm this. I might look for such information if I were into racing, and the "prevailing opinion" probably would be sufficient that I'd use it anyway for racing.

Heat rise in tires even in normal passenger vehicle use can be "noticeable," but except in very rare cases is not sufficient to affect performance, safety, tire life, or fuel economy.

The exception to the significance of heat rise in normal uses is that a significantly underinflated tire can generate enough heating to be dangerous. Using pure nitrogen won't affect the fact that poor maintenance can kill you.

I do know personally a few people who make a habit of drivin' 'cross Texass, 2 or 3 hunnert miles at a trip, when the air temp is 105F, the pavement is at 180F, at speeds of 70 mph (the legal limit) or higher, on the "gimme the cheapest ya' got" tires from Wally World. I've never known one to blow a tire due to heat - except where the tire was "flat" when they left. They replace when the tread looks like a hobo's shoe soles, and the sidewalls resemble the baby that the dingoes got to.

I don't ride with them much (except possibly for local trips and with my teeth (etc.) clamped pretty tight); but their experience under "abusive conditions of ordinary use" sort of says that the heat thing isn't much of a problem in "common use."

(Feel free to discount this "argument" as "anecdotal" and "not worth a shit," 'cause that's what it is.)

Proper maintenance, including inflation, and selection of tires appropriate for your driving habits are your keys to safe and happy motoring. For most people, inflating with nitrogen is just speckled paint and go-fast stripes. If it makes you feel better, that's fine, because it won't hurt you except in your pocket; but it's not on my list.

John