Migration of ethyl group in pinacol rearrangement
The two possible compounds resulting from the pinacol rearrangement of your substrate, 3,4-dimethylhexane-3,4-diol, are as follows.



A quick Reaxys search throws up a paper in which this pinacol rearrangement was investigated.1 I quote directly:


  Four different investigators have studied the pinacol-pinacolone rearrangement of 3,4-dimethyl-3,4-hexanediol. It was finally demonstrated by Nybergh that, contrary to the conclusions reached by the three previous investigators, the rearrangement involved the preferential migration of an ethyl group rather than a methyl group.


The paper describes how the authors carried out the reaction and analysed the mixture to determine the ratio of the two products shown above. In 1956, there was much less technology than we enjoy now, so no NMR and no HPLC. It turns out that they had to hydrogenate the products to make a mixture of the corresponding alcohols and figure out the ratio by IR spectra. The result: $70\mathrm{-}73\%$ of 3-ethyl-3-methylpentan-2-ol (corresponding to the ethyl shift), depending on the form of the starting pinacol used.

Unfortunately, I can't access the (very old) journals that claim that the Me shift happens preferentially.

References

(1) Reeve, W.; Karickhoff, M. The Pinacol Rearrangement of meso- and dl-3,4-Dimethyl-3,4-hexanediol. J. Am. Chem. Soc. 1956, 78 (23), 6053–6054. DOI: 10.1021/ja01604a028.