Flexing upright twists tunings

[Greg Graham]

Ron N wrote, in jest "...speculation on the possibility that we don't 
actually tune pianos, but only the space they occupy, so 
moving them doesn't throw them out of tune so much as 
dislocating them from the tuning, which would theoretically 
remain in the space in which it was installed. Unfortunately, 
no one seems to have tried moving the piano back to the spot 
in which it was last tuned to see if the spacial tuning 
reconnected with the piano."

Well, reality mirrors comedy.

One Yamaha U1 I know will go badly out of tune at the bottom of the plain strings if you move it two feet across the band room floor, only to go back in tune if you move it two feet back.  

In one spot, both front casters are firmly on the ground (a hard, somewhat uneven, concrete floor).  In the other spot, one of the front casters is about 1/8th inch above the floor.  A3 will drift about 12 cents, A4 half that, A2 and A5 don't change.  An octave from wound strings to plain will sound great in one spot, and horrible in the other.  

The question is:  Is there anything that can be done to eliminate the flexing?  

Would dollies under the bottom board work better because the wheels are closer together, and perhaps not so firmly bound to the case?  

I've thought about providing small squares of 1/8th and 1/4 inch plywood for the pianists to shim the hovering front casters as needed.  I'll call them "Tenor Tuners".  $59.95 each, if you call before midnight tonight!

Any other (more practical) ideas?

Greg Graham