The PF made nose-down control inputs and alternately left and right roll inputs. The vertical speed, which had reached 7,000 ft/min, dropped to 700 ft/min and the roll varied between 12 degrees right and 10 degrees left. The speed displayed on the left side increased sharply to 215 kt (Mach 0.68). The airplane was then at an altitude of about 37,500 ft and the recorded angle of attack was around 4 degrees
This is the important part: "The PF made nose-down control inputs and alternately left and right roll inputs." The PF did indeed stick the nose down. It didn't work.
Why do you say it didn't work? Right after the part you highlight, it says: "The vertical speed, which had reached 7,000 ft/min, dropped to 700 ft/min". Seems to me (from the little we are told here) that it WAS working.
Problem is, the very next para goes on to say: "The PF continues to order pitch up" so at some point the PF had evidently gone from giving nose-down inputs back to giving nose-up inputs. Why? This is all within the first minute, so those pitch-down inputs cannot have lasted very long.
Now, the stall horn did go off as the A330 reached 37,500 feet, that is shortly after (at least in the narrative) the pitch down commands. Is this what you mean by "it didn't work?" More to the point, could the PF have interpreted this as indicating that pitch down "didn't work"?
But what is surprising about a stall horn going off when a heavy airplane has just climbed rapidly above the cruise altitude that the very same PF had, just a few minutes earlier, say they could not climb above, probably losing speed in the process, as well as increasing its AOA? It seems to me (admittedly from the comfort of my couch) that the reaction should have been: I haven't solved the problem yet, and need to keep at it.