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Could a high-pressure, low oxygen atmosphere reduce fire risk while still being breathable?

Years ago I read a letter in a magazine suggesting a method for reducing the fire risk in an enclosed environment such as a space shuttle.  The idea was to reduce the oxygen concentration to the point where most common materials won't readily combust, and compensate by increasing the pressure so that the partial pressure of oxygen was the same as it is in "normal" air at sea level, making it breathable.  

The assertion was that the ppO<sub>2</sub> is the deciding factor in oxygen-haemoglobin binding, so a person could live comfortably in such an atmosphere, but the increased pressure would not increase the fire risk.

I understand deep-sea divers sometimes use reduced oxygen concentrations to compensate for high pressure, so the first part  seems reasonable.  However it seems to me that a higher ppO<sub>2</sub> would also make oxygen more readily available for other reactions, including fire, and at least some fires do burn faster at higher pressures. Hence I wouldn't expect much benefit, if any.  I have looked for research on the topic but drawn a blank, which might be a clue about the likely efficacy.

**Would using a high-pressure, low-oxygen atmosphere as described give any useful reduction in fire risk?**