This is too long for a comment, so I'm posting it as an answer.
I though it should be possible to recover tuning of the instrument using harmonics, after all due to how Fourier transform works, higher frequencies have higher precision. I have opened a recording of the last chord of C major prelude (cut out and copied several times from both left and right channels) in spectral imaging software and the harmonics I recovered form the recording do not make sense to me.
For a FFT of size 65536 and 44.1 KHz sampling (which gives accuracy of 0.67 Hz) I got the following buckets (manually picking those with high values):
89, 97, 194, 244, 268, 292, 357, 388, 446, 486, 491, 714, 780, 889, 974, 1072, 1169, 1268, 1365, 1368, 1475, 1563, 1665, 1756, 1765, 1956, 1970, 2065, 2154, 2166, 2267, 2467, 2552, 2751, 2778, 2880, 2941, 2968, 3473, 3543, 3768, 3946, 3983, 4149, 4497, 4761, 5016, 5543, 6618
I know that harmonics of acoustic instruments deviate from the ideal series, and that it can vary in time, but I guessed the mean should be good approximation, and I hoped these differences among different strings will be similar.
Unfortunately my approach didn't work. The differences between harmonics and their ideal counterparts were just too big and unpredictable. Also, there were some larger clusters of frequencies, too large even if you were to count colliding harmonics of different notes. Sometimes good piano tuners introduce slight differences between different strings of the same note to make the sound more rich, but these are usually very small. Also, I didn't have the recording in a lossless format, so perhaps the encoding messed the sound (although I doubt it).
In other words, I have no idea why that didn't work, but the problem of recovering actual tuning seems to me much more difficult. Although the general issue is accuracy, if I were to guess the actual cause looks more like the general complexity of sound of the piano, including interference of all the other strings that were not hit, but still play.
Finally, even if we were able to recover note heights with reasonable accuracy, that might not be any known tuning (e.g. the high frequencies might be a bit more equal-tuned than the rest, while the low frequencies might be just stretched). In fact, if we go into such detail, then each acoustic piano is slightly different (in particular how they resonate), and I would expect that tuners tune pianos so that they sound good rather than just use some particular scheme, even if they try to achieve effect similar to, say, Werckmeister III tuning.