I think the largest issue with horns and disappearing is due to uneven polar response of the horns. The 2nd link has some polar patterns of commercially available speakers in it.
This blog article is the first in a mini-series about speaker directivity or off axis frequency response. For us the two most important things in determining sound quality in a home theater or stereo are the speaker and the room. Above the room’s transition frequency the speaker’s directivity...
www.acousticfrontiers.com
www.princeton.edu
When you have multiple horns that cover the ear's sensitive frequency range without having the same dispersion pattern it's audible because the off-axis sound is not similar to the direct sound. So what could be benign reflections due to longer path length/delay times becomes interference the brain can't process as easily and recognizes as different. This can destroy the perception of the spatial cues in the recording and make the speaker unable to disappear.
This is the reason a speaker like Bill's, Earl Geddes, the multitude of speakers that use a dual diaphragm CD to cover from ~400 Hz on up, disappear better than a vast majority of multi-way (4+ way) or Avantgarde-type designs. The latter tend to have uneven polar patterns. Bill's and my own speaker use wideband driver that can cover the entire frequency spectrum where it matters. Earl Geddes and Duke LeJeune match the polar pattern of the woofer to the CD higher up, maybe 800-1200 Hz, but the resulting polar pattern is even. Dual concentric driver CDs use a single horn, and I'd argue the crossover is a bit of an issue vs using a single wideband driver or diaphragm, but the polar pattern is generally better vs multiple horns.
Another factor that can destroy the ability of a speaker to disappear is diffraction. This can be an issue with all speaker designs, but with horns the horn it's self often causes HOMs or horn honk, and to save on diameter horns are often not given a full round over at the mouth, which causes audible diffraction around the edge of the mouth. Also, less than ideal horn expansion profiles are often used to save on diameter, which causes some HOMs. Finally, the material a horn is made out of matters, just like the material speaker cabinets are made out of matters. So you can see some popular commercial speakers use non-ideal profiles with no round over and cheap injection molded plastic horns. Then add poor polar response on top of that and it's a wonder the speaker can image at all!
Cone 'n' dome box speakers usually suffer from audible diffraction, which is why they sound like box speakers. The diffraction is similar enough the sound signature is recognizable to many. Some manufacturers do a good job with this due to cabinet design, but it's very difficult to avoid entirely. It also explains some of the interesting shapes from Vivid and others, as well as the speakers that use a spherical enclosure as this is considered most ideal.
IME, this is also an issue with the concentric mid/tweeter designs of Andrew Jones, as used in KEF, TAD and ELAC speakers. The midrange forms a horn or waveguide for the tweeter but it's not possible for the midrange cone to be a perfect LeCleach horn shape and the compromise is audible. It is good for other reasons so I'm not saying it's bad... I own a pair of Pioneer S-1EX which use the TAD Mg mid/Be tweeter. They are excellent but only disappear 95%, and I've heard the same from a few other speaker designers. It's always a compromise so it's not wrong, but I think a separate mid/tweeter can disappear better if that's your priority.
It's also an issue with the GPA concentric woofer/horn design, the horn must be compromised so it doesn't cover the entire woofer.