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The premise of SCAN, in canon: the King of Bohemia says Irene will "ruin" him by exposing their past affair. That is, the King's current fiance, a princess, is SO puritanical that she will not marry him if she knows he ever had anything to do with an "adventuress."
Now, I know this is Victorian England,and it was presumed that women were shocked by sexual matters. And no doubt the standards of conduct for men were different than they are today. But I guess I would have expected the usual double standard, with men being allowed to have those kind of "adventures" before marriage (and perhaps during), and the highly-born wives being conditioned to either accept it, and perhaps be grateful not to be "imposed on" for sex, or else regard it as all the slutty adventuress's fault for being available. In which case Irene would damage herself more than she would the King, by saying anything.
There IS a double standard in the sense of "She was so exciting but she's not good enough to marry," (I think the King is still hot for Irene, actually), but the point is, the affair seems to threaten the man's reputation, at least his political position, as well.
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Maybe there´s somethings so shameful connected with Irene´s name in Bohemia, that any connection with her can cast a shadow on a nobles´s name?
(In truth, I think that poor ACD just used first thing that came into his mind and didn´t think very deeply about it. )
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According to Holmes himself: "Your Majesty, as I understand, became entangled with this young person, wrote her some compromising letters, and is now desirous of getting those letters back." The letters were in his handwriting, on the royal notepaper, sealed with the royal seal--and there was also a photo of the two of them, which would argue for the authenticity of the letters.
Presumably there was something more indiscreet than proof of an affair in the letters--perhaps a promise of marriage? Certainly something that would cast the King is a particularly bad light.