My gut instinct says that this should be machine-washable.

If you have the time, space and warm weather, you could leave it soak for a day or two in warm soapy water (so warm that you don't feel the temperature when you put hand in it) and some enzyme-based detergent.

Then leave it out the sun for a day or two to drip dry.

However, since it is valuable and a nuisance to replace, you could try your local dry-cleaner's. They surely have such problems every now and then.