Here's from the demo measurement, which happens to be my own speakers
Three way speakers with 1st order crossover between tweeter and midrange, and the two are time aligned..
And this is how I prefer to look at it: The full amplitude, but over a short time span.
It has some low level pre-ringing, which is a measurement artifact when the measurement fades out (the measuremen is linear phase).
But note the clear leading transient. One up and one down, sometimes another ... is very typical fro a conventional speaker. A horn rig may be a bit more fuzzy, but I would expect something similar in your case. When you zoom in as much as you do it is difficult to see the relative magnitude of the high frequency oscillations. Here's what it looks like when I try to do a somewhat similar zoom to yours:
If I filter it, and allow for a higher amplitude, it looks like this:
Please try to do a zoom like the first one I posted .... and the unfiltered version.