Awesome persimmon tree, it looks very healthy! From what I've read, if it was an Asian type persimmon it won't be astringent, but if it is an American type, it will be astringent before being fully ripe.
I haven't grown persimmons yet personally though, but I am hoping to plant some next year and get some seeds from them to grow!
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Most japanese varieties are actually astringent until fully ripe. So I'm assuming there is a particular gene for astringency that is not expressed in the non astringent ones.
I'm not a master of genetics so my answer is all guesswork, but I would say it's pretty likely the tree grown from seed will have the astringency gene, and will probably express it.