I'll simplify all your lives by saying HOM's are mainly just diffraction related back wave that emits when waveguide profile changes too suddenly and sound diffracts, be it inside the device or at the mouth. The back wave emits backwards where the sound came from, thus back inside to the device, and reflects from inside the device back out.
All this shows up as direction and frequency dependent undulations in frequency response plot. Or group delay plot, as it's sound arriving after direct sound. Now, this shows easily in simulation because the source can be anything ideal so the waveguide contribution to diffraction is clear. If there is more in frequency response measurement it must be from the compression driver, or from junction of the two, and would show up with appropriate delay. So, if you have a waveguide that has proper roll over on the mouth as part of nice continuous profile all the way from phase plug one doesn't see any wiggle in frequency response to any direction up until top octave, where wavelength is smaller than the throat and dominated by what's coming from inside the driver.
If there are some other HOMs in the system, like the waveguide resonating or something, they would not be visible with all the diffraction going on, which is kind of a huge phenomenon in frequency response plots, could be many db. If one makes a diffraction free (relatively) device, and still see wiggle, it should be easy to test and reason what's the issue. If there is no wiggle in frequency response I'd quess there is no HOMs, definitely there is no diffraction then. No matter which profile horn, OS, exponential, no horn at all, if there is no properly sized and shaped mouth rollover there would be a lot of diffraction showing in any plots. Basically, diffraction makes secondary sounds emit from the device, and if HOMs are secondary sounds, I don't know whats the difference in practice, at least they would contribute to similar perception of sound I bet.
As disclaimer, I don't know what HOMs would actualy be, they seem mysterious voodoo like entity but seem awful lot like diffraction. Unlike HOMs seem to be, diffraction is very easy to detect both in measurements and in very simple simulations, it's relatively easy to reason with and happens with any physical objects, also with waveguides.
edit. simplest possible example a "conical" horn
https://www.falstad.com/ripple/Ripple.html?rol=$+3+512+64+0+0+778+0.048828125 s+2+255+21+2+0.46680807839388144+0+10.001413602849373+99.99941507823934+1+0 w+0+207+91+253+18 w+0+306+91+259+17 w+0+253+20+259+20
Same with extension, non smooth profile with sudden change and it emits the backwave as well.
https://www.falstad.com/ripple/Ripple.html?rol=$+3+512+64+0+0+778+0.048828125 s+2+255+21+2+0.46680807839388144+0+10.001413602849373+99.99941507823934+1+0 w+0+207+91+253+18 w+0+306+91+259+17 w+0+253+20+259+20 w+0+365+132+305+90 w+0+208+90+155+131
Click the links to see it happen.
edit2. mabat posted some ripple tank examples on ATH thread:
https://www.diyaudio.com/community/...-design-the-easy-way-ath4.338806/post-6714375