Well, if it's true, is would mean an extensive straw hat production in Milan in this time and probably also export of this article. Which somehow would be logical, as Milan definitely was a center of trade and likely also for the haute couture of the time. Cheap straw hats likely wouldn't be exported.
If it, as you say, had been a relevant factor in the issue of the siege in 1450, it would even give a logical reason for Sforza to have a straw hat at the Magician's table.
If Milan had been a big factor in the contemporary production, it seems plausible, that Bianca Maria's stay in Fermo and in Cremona caused, that these regions developed just this item as a local speciality. Such a context would even give some background for a local anecdote in Fermo, which might be not literally true, but just transforming the message, that the region got into the straw hat business cause Francesco Sforza and Bianca Maria married and had some time in Fermo (filling the small town with interests in items, which were used for a noble court).
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... perhaps you should steal your brothers book.