With my QCX-40 is a data point, the answer is yes. I did a quick measurement of my rigs output power (scope method) at half volt steps from 14.5 to 7 volts supply. 0.5 Watts out was at 7.2 VDC supply. Spreadsheet attached in .xls and .ods (LibreOffice Calc) formats.
Again, this is only the measurement of one example of the QCX and mine seems to run a bit low on output power. Maybe a 1 turn mistake on inductor winding, but that's fine with me.
For WSPR I built a cheap and simple 30dB attenuator and left my normal supply voltage at 12 VDC (2.5 Watts).
Brent DeWitt, AB1LF