Hi Dave
I clean silk rugs with out too many problems,
as you realised it's all about low moisture as its a natural fibre you have to be careful what you are using on it.
Very good vacuum first in all instances followed by one of the methods below,
I clean using a dry clean solution like prochem Dri Pro applied by spray to surface and a light mist over a pad, I then roto all over, change pad and roto again, finishing with pile re-alignment.
If more heavily soiled I would Texatherm.
If very heavy, apply solutions no4, agitate in and rinse with CFR low moisture tool, re-align pile and fan assist the drying,
When dry spray with dry clean solution and pad/roto,this will soften pile nicely and then re-align.
Ps if you do not realign pile it will set in the positiion you last left it and restoring it back can be difficult and some times impossible.
If unsure, don't do it as this is a high value item and not something you should be learning on.