Thank you for providing me the reference.
A burning test of this kind is not in any way capable to tell the difference between a mixture and a reaction product, The main reason is that it is a destructive test and thus does not allow you to say anything about the formation of new bonds. The way to follow the reaction is by studying the absorption infrared spectrum which will show the disappearance of the OH bonds and the formation of the B-O-C bonds. No doubt this has been done and probably published.
A quick check of the literature shows the following for the methyl analogue.
One gram of boric acid dissolves in 6 ml of boiling alcohol, which means that it does not react.
The tri ester can be made by the reaction of boric acid and methanol under certain conditions described in Schlesinger. J. Am. Chem. Soc. 75,213(1953).
Several other processes are described in six U. S. patents. I can give you the numbers,if you are interested, but what it indicates is that the simple mixing of boric acid and methanol does not give the esters, because patents would not be granted for such a simple mixing procedure.