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

Couple of things come to mind. First, it could be deliberate to enhance imaging. Next, it may not be "real" in the sense that the overall energy you hear in a room may be different, or it is designed for zero toe in and listening at an angle which eliminates this issue.

Despite the guys at ASR getting their panties all knotted up, if a bear poops in the woods and there’s no one there to smell it..... oh, I forgot the metaphor but basically if you don’t hear it who cares?

This from a speaker maker who sweats each Hz and each dB!! 😂

Might be fun to MEASURE it in your room, and them switch a compensating parametric EQ curve in and out to see if you care and which you prefer.

This is kind of tickling my memory of theater speakers, which were often 2-way and horn loaded.  We never had perfect crossover matching, but we did have very good sounding theaters.  This horn/woofer alignment striikes me as similar.

One other thing that I'm thinking about is that ASR is measuring the speaker at 1 watt.  Lots of speakers measure close to flat ONLY at 1  watt and then have their performance rapidly degrade when louder or heated up by playing (i.e. thermal compression). For a speaker like this, excellent performance even when played hot would matter more for me.

Ok, another thing, there are some crossover alignments which forego absolute perfect amplitude for excellent impulse response, the name of the crossover alignment escapes me, but not impossible this is an example of it.

If you want to know exactly what the crossover characteristics are, it’s quite easily done. Get a used audio frequency generator and oscilloscope and plot the curve with some graph paper or Excel.

@erik_squires I think you were thinking of Linkeitz-Riley crossovers which are nominally -6dB at the crossover points.

And as far as the ASR crowd, I have a sneaking suspicion the individuals commenting have never heard the JBL 4349 or heard of Linkeitz-Riley filters, let alone the theory behind them.