Proceed with caution
I've used a variety of fabric and upholstery cleaners with varying degrees of success, but if your chair has original Girard fabric on it (and therefore the associated value) I'd be very careful and consider taking it to a pro.
In addition to the foams and sprays I've tried, I've had a shop with a cleaning machine and correct cleaners for upholstery do some pieces for me and they always came out great.
Maybe let your fingers do the walking and see what a good shop would charge you to clean it properly before you buy cleaners and take it on yourself.
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