It doesn't really say if the PIO did involve those 4 touchdowns. In
that case it sounds to me like the classic problem of the tail going
down due to inertia, increasing AOA again and starting to fly because
of the increased lift.
In this cased using airbrakes will stop the PIO more quickly because
it is less likely that the increased AOA will be enough to make the
plane fly again. Also, the number of times this happens will probably
be less since you bleed speed more quickly, which also reduces the
amount of lift you're able to produce at the max AOA.

On a side-note: I am interested if anyone has ever performed Cl-Alpha
measurements on a glider with airbrakes applied?