The pitch down with full flaps went away with the larger vertical stabilizer on the K model, according the the author of "Cessna, Wings for the World". The author was William Thompson who spent spent 28 years at Cessna as an engineering test pilot and Manager of Flight Test & Aerodynamics.
Actually, it was the "L" model; the "K" as the last model that had the pitch down problem, and it was the dorsal fin that fixed the problem, rather than merely the vertical stabilizer. Here's the entire excerpt:
With the advent of the large slotted flaps in the C-170, C-180, and C-172 we encountered a nose down pitch in forward slips with the wing flaps deflected. In some cases it was severe enough to lift the pilot against his seat belt if he was slow in checking the motion. For this reason a caution note was placed in most of the owner's manuals under "Landings" reading "Slips should be avoided with flap settings greater than 30° due to a downward pitch encountered under certain combinations of airspeed, side-slip angle, and center of gravity loadings". Since wing-low drift correction in cross-wind landings is normally performed with a minimum flap setting (for better rudder control) this limitation did not apply to that maneuver. The cause of the pitching motion is the transition of a strong wing downwash over the tail in straight flight to a lessened downwash angle over part of the horizontal tail caused by the influence of a relative "upwash increment" from the upturned aileron in slipping flight. Although not stated in the owner's manuals, we privately encouraged flight instructors to explore these effects at high altitude, and to pass on the information to their students.
This phenomenon was elusive and sometimes hard to duplicate, but it was thought that a pilot should be aware of its existence and know how to counter-act it if it occurs close to the ground.
When the larger dorsal fin was adopted in the 1972 C-172L, this side-slip pitch phenomenon was eliminated, but the cautionary placard was retained. In the higher-powered C-172P and C-R172 the placard was applicable to a mild pitch "pumping" motion resulting from flap outboard-end vortex impingement on the horizontal tail at some combinations of side-slip angle, power, and airspeed.
However, the book also says:
To eliminate the previously mentioned "pitch pumping" in flaps-down sideslips, the 40 degree flap setting was reduced to 30 degrees on the C-172RG, C-172P, and C-172Q.
