That's what I would say, although I haven't done that exact comparison.Try plugging in a 16 ohm cabinet vs a 4 ohm cabinet with the same exact speakers in them, with your amp impedance selector set correctly for each. Are you saying the 16 ohm cabinet will not be lower volume than the 4 ohm cabinet?
It's not really possible for versions of the same model speaker with different impedances to be exactly identical, but their sensitivity ratings are measured with the voltage applied adjusted for the impedance, so if the sensitivity ratings are the same, they should produce the same SPL when matched to the proper tap. Tube amps deliver maximum power when the load is matched to the tap, and they shouldn't provide any less power when using the 16-ohm speaker with the 16-ohm tap than they do when using a 4-ohm speaker with the 4-ohm tap.
I've not heard any difference like what you've described when I've compared 16- and 8- ohm speakers, but I've never tried with 16- and 4-ohm speakers. Of course the impedance ratings are nominal, and in a particular case it might be true that one of the cabs is actually better matched to the tap than another, but I suspect that would be a rather modest factor at most, and it likely wouldn't represent a general rule.
Now if we we're talking about solid state amps, using a 4-ohm speaker would definitely generate more volume than using a 16-ohm speaker. People with a lot of experience with solid-state amps -- guitar amps or otherwise -- could potentially suffer from confirmation bias when doing non-blind comparisons with tube amps.