Some quite respectable newspapers, for example, appear to have got the impression that the pilot caused the initial 'steep climb.' Even the BEA thing doesn't actually say that - indeed, in very 'spare' wording, it rather confirms the opposite, that the pilot countered the climb with 'nosedown control inputs' and restored the aeroplane to a situation of reasonable airspeed and a reasonable AoA:-
If you look at the paragraph preceding your selective quote (timestamp 11s earlier), you will find:
The airplane began to roll to the right and the PF made a left nose-up input. The stall warning sounded twice in a row. The recorded parameters show a sharp fall from about 275 kt to 60 kt in the speed displayed on the left primary flight display (PFD), then a few moments later in the speed displayed on the integrated standby instrument system (ISIS).
I concur that we don't know enough about the nature of that input to conclude with certaincy it caused the climb, but there's nothing in that note saying it didn't. So far the chain of events NU input by PF -> climb -> ND input -> reduced climb is the least complicated (and thus probably a likely) way to connect the few dots we have.