Using HHs for MM IB jump to PE

I work a decent MM shop in a good but slightly niche group and I've had some PE head hunters reach out already regarding a few openings. However, each role is not located in my city (Chicago) or operates in my desired sector(s), and I am not open to relocating any time in the future and I don't really have an interest in doing something far different than my current sector, nor do I believe a PE firm specializing in industrials, for example, will have interest in someone with expertise in a completely different sector.

With all that said, do you think it's worth it to reach out directly to some of the HH firms out there and submit my resume? I've seen a lot of users here recounting their experience of getting shut down by HHs since they didn't have BB experience and I would prefer to know if I should even waste my breath with that process or if I should go the non-traditional route and try to slowly build PE connections through my network and try to line something up with a larger time horizon.

I'd prefer LMM - MM funds (generally less than $300 - 400M in average check size), so I'm not gunning for a MF or a top tier UMM here.

Any and all advice is appreciated.