The INV-HG mkIII comes standard with a 68-shore Duo-Lyre. But this is actually much more stiff than you would expect for that shore rating because of increased stiffness in certain areas compared to the separate Lyres such as those used in the INV-6 and INV-7.
See this thread for excerpts of my correspondence with Simon Davies at Rycote where he explains:
http://taperssection.com/index.php?topic=180990
Oh wow, I didn't even know that I had the dual lyre mount. LOL. I understand his explanation perfectly and it sounds like what I need is a pair of 62 shore single lyres. Thank you!
Hold on. I'm looking at the mics in your signature, and for any of those, you definitely do NOT want the 62 shore lyres. I have those for my CM3s because they are so lightweight. The 62 are so compliant (floppy) though, that I can only use them for the individual mics, and only if the cables are well dressed. When using a stereo bar, I need the stiffer lyres. (I'm actually using the 68 INV-HG mkIII for that now, but as that's too far in the other direction, I'm going to swap it out for the 72 shore single lyres which should be perfect.)
For Schoeps or any "normal" pencil mics, I would say you want the 72 shore lyres, which are the ones that come standard on the INV-6 and INV-7 mounts. For a stereo bar of mics like yours, you might even need to go to the 82 single lyres though I have no direct experience with them. I believe Bean and a few others here use them for that purpose, so you could ask them, or email Rycote directly who as you have seen are very helpful.
The 72 lyres are in stock at Adorama right now, if you have a base to mount them to. They are a special order everywhere else I've looked. Otherwise, the complete INV-6 and INV-7 are in stock at all of the usual places.