Just anecdotally, I've thought the sound of push pull tube amps is in the transformers mostly. I've had a few tube pre's back when people had the idea you could get much of tube sound by using a tube pre into a ss power amp. I didn't find it to be the case. Good tube preamps didn't really have a sound. Most put out more voltage than necessary, and sometimes adjusting gain was a good step, but they were not noticeably different than good clean SS or FET preamps.
Now I've done some careful comparisons of tube power amps, but all sighted. Discount that as you see fit, but I found Ultralinear tube amps to have a similar and non-transparent sound, triode push-pulls had a sound (which was my favorite) and SET's have a sound (though these varied more than the the others). It was most apparent when I did series tests where I put a power amp with loaded output between the source and the power amps driving the speakers. I could then switch the power amp under test in and out of the circuit. I found there were FR differences, but also they seemed to create a 3d space, good ones seemed smoother, and usually seemed to punch up subjective dynamics. The FR differences were real and easily measured. The creation of space, smoothness and dynamics not so easily measured. Other gear with transformers often seems to have a smoothness, and ability to add some spaciousness to sound. Not always does it add the dynamics, so maybe more than just the transformers did that or maybe at power amp levels where saturation can be an issue it creates false dynamics. I've thought a simple push-pull triode preamp using 12ax7s or 12at7's which fed a small output transformer might be a good thing to do. Basically a mini push pull triode amp to pre-color the signal fed onto your power amps. Tim's E.A.R. preamps used transformer coupling at the output.
What I found is that good SS amps can fully and faithfully re-create all that spacious, dynamic and smooth sound of a triode power amp. Because if you put a triode amp in front of the SS amp you get the full triode sound.
But none of this answers definitively the OP's question.