Yours, and the reason posted by Lonewoolf - no additional throttle during the climb - and above all - perhaps not a surprise to you - the energy conservation calculation showing that the kinetic energy at FL350 and FL375, based on the BEA text are matching the climb potential energy, showing that no additional power was applied, are convincing.
Really?
A simple conservation of energy calculation would show that a 3000ft climb requires a loss of 260 Knots of true airspeed.
gH = 1/2 x (V^2) where V is the speed change and H is the height change
So in metric units V=SQRT(2 x 9.8 x 914)
V=133 metres per second (approx 260 Knots decrease)
And that assumes a 100% efficient energy transfer - not usually provided by an aerofoil heading deep into the back of the drag curve.
So I would not be so sure that there was not a significant input of lift from the surrounding environment.