How widespread is labialization of [ʃ] and [ʒ] in 'dish' and 'beige'?
Solution 1:
This article implies that labialization is universal in English for those sounds, and cites a number of authoritative-sounding references (which you might want to investigate) – and I've been unable to locate any contrary views. On the other hand, if you yourself have found otherwise, you may be on the verge of a major upset to the conventional wisdom on the subject.
But note the distinction made in the article between "tight rounded (/w/)" and "slight rounded (/ʃ/, /ʒ/, /tʃ/, /dʒ/, initial /r/)". It could be that you're just seeing a gradient, not a complete absence of rounding.
Apparently in Tillamook a rounded sound is created with a cupping of the tongue instead of a rounding of the lips, so perhaps it's not inconceivable, but that doesn't seem to be what's going on in English.
Regardless of the anecdotal support for your proposition in the comments here, you might want to post your videos to be examined by others as to the amount of lip-rounding, if any, to see if it might be worth going further in your investigation.
Admittedly, I myself can do it both ways, but completely non-rounded seems very forced and unnatural (which of course is as anecdotal and unobjective as the other personal comments here).
Of course I may be overthinking my own pronunciation, and your subjects may be overthinking theirs, which is just to say that it would be a complicated matter to answer exactly how common the non-rounding phenomenon is (or even whether it exists at all), probably requiring a scientific (perhaps double-blind, and certainly cross-population) study that didn't focus the participants' minds on the particular sounds of interest, potentially influencing the results.
For example, you might want to take video of them saying a list of words to which those two, dish and beige, (or similar words) seemed to be random additions. And you might want to add additional parts to the study, with different parameters – e.g. an exercise in which those sounds were embedded in full sentences, with different sounds coming before and after them – to see if that made a difference.
And that study would presumably have to have peer-reviewed results, with the video available to be assessed by objective third-party observers.
I await the results!