This has partly been answered to an accuracy of about 1% in the other thread. People have suggested a value of about 1.25
From a theoretic angle, treating H2O as an ideal polyatomic (nonlinear) gas at 3000K (a very good approximation, if you ask me), Cp and Cv values would depend on whether or not you are exciting the vibrational moles in the molecule.
If this temperature is too low (which I doubt, since you typically start to see signs of exciting vibrational modes at about 1000K) to excite vibrational modes, then the molecule has 6 degrees of freedom, and thus Cv = 3R and Cp = 4R, making Y = Cp/Cv = 1.33
If this temperature is high enough (I think this is almost certainly true) to excite both of the dominant vibrational modes of H2O (one where the H atoms have opposite velocities relative to the O atom and the other, where they have the same velocity) then the molecule has 8 degrees of freedom, making Cv = 4R and Cp = 5R, and hence Y = Cp/Cv = 5/4 = 1.25
If you also include a flapping mode (I'm not sure what the activation temperature for this mode would be - just higher than the other modes), then you have Y = 5.5/4.5 = 1.22