Years ago, when I owned a silk shirt, I used to hand wash it and let it drip dry therefore it hardly needed to see the iron.
I'd be tempted to try that first, if the colour doesn't 'straighten out' then try pressing it with a cotton cloth covering it rather than the iron directly over the silk.
If all else fails, would it be possible to remove the upper part of the collar to create a collar band?
Charity Shops Making Donations Uncomfortable
