Currently, only the GI, DAI, Batguard, GT, and Talon have been sub-20'd (note that some of those aren't in the plat solo HoF but can be found on Youtube) to my knowledge. The Kroguard could also do it if someone cared enough to grind it out (currently I believe the fastest Kroguard time is 20:xx). I think it's remotely possible for the Reegar AIU, but even then I'm a little skeptical.
Beyond that, I doubt many kits -- or any at all -- can be sub-20'd. On paper it looks as if someone could put a Reegar on, say, the Valkyrie and get a sub-20 completion (currently the fastest time for her is 21:36 with the Venom on Glacier with AC V), but in practice I feel she simply fails to tank enough damage as she closes into reegar range with the enemy and maintain enough sustained DPS to reach Reegar Batguard/Kroguard levels. Maybe I'm just bad and die a lot tho.
Not to mention, of the three maps that have been sub20'd to my knowledge (White, Giant, Glacier), the fastest one (Glacier) isn't particularly grenade rich. I think that effectively knocks a lot of the powerful grenade-reliant kits out of competition.
So in my opinion, it's impossible for all kits, and improbable without an essentially perfect game* for even some of the most powerful kits that haven't been sub-20'd. There are better players and more versatile soloists than I who might have a differing opinion, but even then, I'd be really skeptical if anyone would even be willing to try such a thing given the grind involved.
* and by perfect I mean pretty much 10 WS, and perfectly placed spawns and objectives and 100% optimal missile use. Which is something I've actually never seen or achieved.