I have not tried it, but here's what I read on how to remove any astringency from these varieties.
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Rojo Brilliante is a persimmon variety with a distinct elongated dome-shape and a wonderfully smooth, bright orange exterior. It can have the astringency removed by carbon dioxide treatment, and is therefore delicious when eaten firm or soft, peel on or peel off. High levels of sweetness and flavour and a thin skin make this a very attractive variety.
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Since the ones from the store is not astringent, I'm assuming it was already treated using this method.