Cleaning chair webbing (Bruno Mathsson Eva, Aalto, etc.)  

Hi folks,

I looked around and couldn't seem to find an answer to this despite my best efforts. Is there a way to clean the linen or hemp webbing on Mathsson Eva chairs or chairs like the Aalto 611 when they get a bit dingy from repeated use? Would really prefer to avoid a full on replacement for the sake of keeping things original if the webbing is just dirty and not damaged. Also to save on the cost of putting new webbing on. I suspect many others with these chairs would feel the same way, so I was surprised to see nothing about this.

My first thought was using a carpet cleaning type wet vacuum, but I fear that'd damage the wood on the areas where the webbing and wood overlap.

Thanks!