It is common knowledge that a poorly ventilated combustion device emits copious amounts of products of incomplete combustion (PICs) that include CO and PAHs. The dynamics of CO2 conversion during combustion processes is debatable in some circles.
The answer is 'yes' provided the temperature is high enough. The negative change in enthalpy is the same as that gained when burning CO to CO2, about 24 MJ/kg.
The phenomenon is seen easily in a low power, hot stove charcoal fire (near the end of a burn) when the carbon on the surface is reacting directly to CO2, not evolving as a CO gas and then burning to CO2 higher up.
In a hot high-H2 environment such as when you have flaming pyrolysis burning coal, the H2 will also strip an O from CO2 because it is highly reactive. This is often overlooked.
CO2+heat+H2 -> CO + H2O
That is endothermic by (thumbsuck) >10 MJ/kg (?) I didn't work out the details. Remember that the water gas shift reaction will decompose the H2O under the right conditions (high excess air, wet fuel loaded onto an existing fire). Heating wet fuel with poor flaming can result in an oxygen level in the stack that is above ambient even though there is a fire burning! I have measured 24% O2.
The Boudouard reaction, named after Octave Leopold Boudouard, is the redox reaction of a chemical equilibrium mixture of carbon monoxide and carbon dioxide at a given temperature.
Temperature plays crucial roll in this reaction, increasing temp. increases CO %.