Help identify this Constellation!

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Not quite, 004 is powered by 561 and 010 is powered by 564 quick set.

That was what I was getting at, poorly, in retrospect.
 
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Finally got it opened! 168.004/14 SC

I still need to find the original strap and tang buckle dimensions, though (I'm assuming it was solid gold as well?)
 
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Strap likely 18/16mm
An 18k gold buckle will probably cost you 拢300 - if you can find one.
Not worth it IMHO
Better to find a period correct plated one.
Need to sort out that bit of rotor rub