The prediction was confirmed in a satellite experiment by Lipa et al., published under the title "Specific heat of liquid helium in zero gravity very near the lambda point" in Phys. Rev. B 68, 174518 (2003).
Since I had to
tell the story repeatedly, most recently for the French edition of Science,
I have set up a web page with all the info:
http://www.physik.fu-berlin.de/~kleinert/feynman/
Enjoy!
Hagen Kleinert
Institut fuer Theoretische Physik
Freie Universitaet Berlin