3-4 dB dip at crossover region: what should I listen for to hear it?

I don’t think you will hear it. I have a pair of Klispch RP8000f (own a pair of JBL 4367s too) and they RP8000f has the same kind of thing at the crossover but much worse on axis. Once in room I really don’t “hear” the dip much. In the Klispch it comes across as a slight lack in female vocal presents.

You can probably effect it with toe and distance from the side walls. If you have a computer for a source just boost that frequency with EQ and see how it sounds.

I think JBL should have used a 2” driver and crossed over lower. I know they prototyped a 2” model. Maybe it was too late or cost too much to hit the price point of the 4349. I have never heard them but the 3” driver in the 4367 is pretty amazing. Based on the white paper the duel ring diaphragms have less break up at high frequency compared to the 4” titanium drivers but have more distortion at the lower end and probably why they crossed over the 1.5” driver in the 4349 so high. I know Greg Timbers said the new driver was more about cost control than performance vs the 4” metal domes and he thought the end result was about the same. The pro side that they D2 driver was designed for was all about power handling and most the pro speakers that use the D2 are 3 ways with mids so the driver is not pushed as low.

on a side note, I see a comment about horn coloration. This to me is caused defraction back down the feed throat which the 4349 should have nearly zero due to its advanced design (computer aided).

White paper here if anyone is interested.
https://www.audioheritage.org/vbulletin/attachment.php?attachmentid=83563&d=1552949127

I would be interested in a dedicated threat about the 4349s as you get used to them and compare them to your other speakers. I am thinking about buying 4 of them for surrounds for use with my 4367s. So the “audiophile” sound quality is not critical for what I plan to use them for I still enjoy reading opinions. 
 

I know I have liked my 4367s enough they have changed my opinion on what is important for hifi. Dynamics have moved from important to most critical. Interestingly I notice it most on subpar recordings. Old kind of cheesy music like Devo and B-52s have punch and snap that I never knew was there. Van Halen jump (the original not the remastered trash) has well, jump. While audiophile recordings like Vanessa Fernandez “here but I’m Gone” snare drum hits are pretty stunning. 

OP: To be clear, your speakers don’t exhibit what I'd normally call horn colorations.  While not perfect, the overall response in the horn region is pretty smooth compared to say an old-school Altec. The complaint ASR makes is that the two drivers don’t blend perfectly at the crossover.

Horn colorations are the peaks and dips in the usable range of the horn, not how they mesh with the other drivers.

Any speaker or combinations of them will have their own peaks & dips, it's own response to an input....it's 'voicing' can either be a 'love it' or eq it to 'make it sound OK' in ones' space....

...and the graphs will reflect hat.....not that the unit will sound 'better' or 'worse', it just shifts the perception to the listener v. the means of measuring of such.

I've messed about with various 'n sundry....and you can only 'tweak' so much before the basic 'characteristics' of a speaker in a very subtle fashion until it turns to muck....mho, of course....

@erik_squires, thanks, that makes sense.

Funny you mention Altecs -- I don’t have much experience with horn speakers other than some old Altec Lancings my friend runs with a pair of MC30s. The 4349s remind me of those speakers more than any others I’ve heard. My friend’s system was my first exposure to hifi beyond the Bose 301s and Sony Receiver I had at the time, so its sound is engrained in my memory.

@james633, yeah, I think I'll start a thread for the 4367s and 4349s -- I'd  be interested in reading about both models.