Woxman
Don't think the pitching oscillations you saw with the Sabre were PIOs. The Sabre had small break out forces - that is you had to push or pull a small force before it would move. This caused beginners to over control each way. Usually mastered after the first flight.
Expect that the PIOs have a relationship with stick force per g which for a Sabre decreased with increase in IAS. That is the stick, having force against a constant spring, became quite sensitive at the higher IASs.
Hard to fix with that system although I would have liked to try a counterweight in the elevator input. Pilots became aware of the hazard and could generally avoid it so like aileron jack stalling at high IMNs they had to grow to accept such little imperfections.
The Gnat was smoothed out by changes to the stick to slab elevator gearing/relationship/ratio.