If you look at the mode shapes of the cone in the link you posted, they would appear to be multiples of quarter waves. A maximum (or close enough to a maximum) is seen at the large open end, if it were a closed end then half wave action would be seen and the plots would exhibit a null at the large end.
For me a horn is a device that increases the efficiency of the driver over a broad band of frequencies, this is accomplished by better transfer of energy into the room instead of reflecting it back into the expanding enclosure geometry. This means that the acoustic impedance at the open end needs to be primarily resistive at the frequencies of interest, this is determiend by mouth size and shape. If efficient energy transfer exists, then standing waves are suppressed (damped).
For the BIB design, the mouth is small compared to the bass frequencies that are being reproduced even if it stands in a corner. This means that the pipe is resonating and the resulting efficient bass output occurs only over a narrow bandwidth, like an unstuffed TL. You will still get harmonics of the fundamental that will produce efficient SPL bands emitting from the mouth of the BIB pipe. At some higher frequency the mouth impedance will become resistive and according to my proposed definition horn like behavior will result. Most back loaded horn designs work in this way, TL action at low frequencies transitioning to horn like behavior at higher frequencies (say between 200 and 300 Hz) and then using a coupling chamber to roll off the back loaded horn's output.
Repeating my definition, a horn is a device that increases the output efficiency of a driver over a broad range of frequencies, the requirement for this to occur is an adequately sized mouth. If the mouth is too small energy is reflected back towards the driver and standing wave resonances will exist. Standing wave resonances are easily seen in the impedance plot of the speaker, multiple peaks mean multiple resonances. A single well damped peak at the tuning frequency of the system is a good indication of a well designed horn. That is my opinion and definition, a maker of a popular commercial back loaded horn does not agree and will loudly tell you flat out I am full of $hit, so be it. But no matter which definition suits you, in all cases the standing waves are quarter wavelength in shape.
