Not unless they have:
1. A submersible IP gauge
2. A place to plug it in without disabling the inflator
3. A very techno-geek attitude that would result in having more interest in your reg's IP than the dive itself.
Now, if someone did manage to check off those requirements and dive with an IP gauge connected, they'd get some interesting info. First, absolute IP rises so that it maintains a constant level above ambient pressure, so the gauge 'should' show, for example, 130 at the surface, 145 at 33ft, 160 at 66ft, etc.....but I'm not sure if a regular IP gauge reads pressure in relation to ambient. Interesting to see how that would go, I suspect someone else would have a better idea than I. Another interesting thing to see would be the amount of IP drop and rate of recovery as depth increases. The 1st stage is working harder to maintain flow at the higher IP and ambient pressures as depth increases. My guess is that a entry level, lower performing 1st would start to show more serious IP drop during demand and slower recovery. I bet a high performing balanced piston like the MK20/25 or atomic would show essentially zero difference in IP drop at anything like recreational depths.
Since you're an engineer, I nominate you to try this experiment and video record it!